f  & 

e=       -o 

II 


|      I 

I  i 


e 

i  1 


^  o 

rri  — * 

i  i 

£  i 


"WHIWHO*5"         '%1»SOV^       ^/^AINOJ^        %OJI1V3-JO^      ^/OJITVD-JC 
;OFCAl!FO^          ^EUNIVERS/^       ^IOS-ANCEI%,        ^OF-CAllFO«fc,       ^-OF-CAIIFOJ! 

^1    ll^TS  l/Or-l    t<US>it  I(US> 


!  ? 

i    s 


^OJIIVD-JO^        %13DNVS01^       %d3AINf!3W 
^OF-CAIIFO^         ^EUNIVERS^        .^lOS-ANCElf, 

^  .v  "%a3AINn-3i< 


HIBRARYOc, 


I 

%fOJIW3-JO 


?     = 

g^j^/|  5 

^unwv-^m^      1 


I    1 


g  g  ijr~  g    g 

I  1 1%  I    I 


%roan^   ^ 


f*i  «*7 

I      I 


AN    INTRODUCTION 


TO 


NATURAL  PHILOSOPHY 


DESIGNED  AS  A  TEXT-BOOK  IN 


PHYSICS 

FOE   THE   USE   OF   STUDENTS  IN   COLLEGE 


BY 

DENISON    OLMSTED,    LL.D. 

REVISED    BY 

E.  S.   SNELL,   LL.D.,   AND  K.   G.   KEMBALL,   PH.D. 

FOURTH  REVISED   EDITION 

BY   SAMUEL    SHELDON,    PH.D.   (WCRZBUEG) 

TROFESSOR  OF  PHYSICS  AND  ELECTRICAL  ENGINEERING,  POLYTECHNIC  INSTITUTE  OF  BROOKLYN 


NEW  YOKE 
CHARLES    COLLINS,    PUBLISHEE 

AND 

THE  BAKER  &  TAYLOR  CO.,  740  BROADWAY 


OLMSTED'S   COLLEGE   PHILOSOPHY 


COPYRIGHT,  1844 
BY  DENISON   OLMSTED 

COPYRIGHT,  1860,  1870,  1832 
BY  JULIA   M.    OLMSTED 

FOURTH:  REVISED  EDITION 

COPYRIGHT,  1891 
BY  CHARLES   COLLINS 


PREFACE. 


r 


"T  is  now  nine  years  since  Olmsted's  Natural  Philosophy  was 
last  revised.  In  this  time  many  changes  have  been  made  in 
the  technical  nomenclature  of  Physics,  many  improvements  in  the 
methods  of  presentation  of  complicated  portions  of  the  science 
have  been  published  by  experienced  educators,  and,  above  all,  the 
whole  subject  of  Electricity  and  Magnetism  has  outgrown  its 
former  apparel.  For  the  present  revision  the  whole  book  has 
been  carefully  gone  over.  A  comparison  between  the  respective 
Tables  of  Contents  of  the  old  and  new  editions  will  indicate  the 
thoroughness  of  the  work.  The  chief  efforts  of  the  revising  editor 
have  been  spent,  however,  in  rewriting  the  parts  treating  of  Elec- 
tricity and  Magnetism. 

The  fact  that  the  electrical  units  are  based  upon  the  C.  G.  S.  units 
has  necessitated  the  introduction  of  the  latter  in  the  Mechanics. 
For  the  sake  of  simplicity,  these  units  have  been  also  introduced 
^      into   the   treatment   of   Specific   Gravities.     The   subjects   Foi'ce, 
^^^    Energy,  and  Work  in  Part  L,  Wave  Motions  in  Part  II.,  Organ 
VM  Pipes  in  Part  IV.,  Spectrum  Analysis  and  Interference  of  Light 
^     Waves  in  Part  V.,  have  been  almost  entirely  rewritten.     A  simple 
^      discussion  of  Lens  Images  has  been  inserted  in  place  of  the  more 
complex,  one  of  the  last  revision.     An  explanation  of  the  cause  of 
the  colors,  yielded  by  double-refracting  crystals,  has  been  intro- 
duced in  the  treatment  of  Polarized  Light.     Additions  have  also 
been  made  to  the  chapter  on  Elasticity. 

Under  the  subject  of  Heat,  the  chief  alterations  have  resulted 
from  the  introduction  of  the  Gram-Calorie  as  the  unit  of  heat. 

In  writing  the  Electricity  and  Magnetism  it  has  been  the 
aim  of  the  author  to  present  clearly  the  principles  of  the  sub- 
ject as  they  are  viewed  at  the  present  time.  To  do  this  in  the 
limits  of  102  pages  has  necessitated  the  omission  of  descriptions 
of  many  experiments  which  are  valuable  only  as  they  happen 
to  correspond  with  instruments  occupying  space  in  many  phys- 


20652 


iv  PREFACE. 

ical  cabinets.  Extended  description  of  apparatus  has  been 
avoided.  A  few  striking  experiments  have  been  described,  but 
the  choice  of  demonstration  has  been  left  largely  to  the  in- 
structor or  professor  in  charge.  Many  new  drawings  have  been 
made.  These  are  chiefly  in  outline,  as  being  thus  best  able  to 
perform  their  functions  in  a  text-book. 

POLYTECHNIC  INSTITUTE  OF  BROOKLYN, 
June,  1891. 


PUBLISHER'S    NOTE. 

QIHELDON'S  ELECTKICITY.  Chapters  on  Electricity,  an  in- 
^— '  troductory  text-book  for  students  in  college,  by  Samuel 
Sheldon,  Ph.D.  (Wiirzburg),  Professor  of  Physics  and  Electrical 
Engineering,  Polytechnic  Institute  of  Brooklyn. 

These  chapters,  prepared  for  and  included  in  the  Fourth  Re- 
vised Edition  of  Olinsted's  College  Philosophy,  are  also  published 
in  a  separate  volume  of  102  pages,  octavo,  cloth.  Price,  $1.25. 


CONTENTS. 


(The  figures  refer  to  the  pages.) 

INTRODUCTION. 

Definition  of  Physics— Definitions  Relating  to  Matter,  1 ;  Properties  of  Mat- 
ter, 2  ;  The  Absolute  Units,  8. 

PART    L  — MECHANICS. 

CHAPTER    I. 

MOTION     AND     FORCE. 

Classification  of  Motions— Uniform  Motion— Questions  on  Uniform  Mo- 
tion, 4  ;  Momentum — Questions  on  Momentum — Force,  5  ;  Motions 
Produced  by  Force — Measure  of  Force,  6  ;  The  Three  Laws  of  Mo- 
tion, 8  ;  Force  of  Gravity — Relation  of  Gravity  and  Mass — Relation  of 
Gravity  and  Distance,  9  ;  Gravity  within  a  Hollow  Sphere,  10  ;  Gravity 
within  a  Solid  Sphere,  11  ;  Questions  for.Practice,  12. 

CHAPTER    II. 

VARIABLE     MOTION. -WORK. —ENERGY. 

Relation  of  Time  and  Acquired  Velocity  —  Space  Passed  Over  —  Space 
Described  during  1st  Second,  13;  Space  Described  during  any  Second 
— Relations  of  Time,  Space,  and  Acquired  Velocity,  14 ;  Laws  of  Uni- 
formly Accelerated  Motion — Formulae — Applications  of  the  Formulae, 
15  ;  Uniform  and  Uniformly  Varied  Motion  Combined — Uniformly 
Retarded  Motion — Space  in  any  Given  Second  or  Seconds  of  Fall,  16  ; 
Questions  on  Falling  Bodies.'lT  ;  Atwood's  Machine,  19;  Work,  20; 
Energy,  21  ;  Transmutation  of  Energy — Calculation  of  Energy,  22  ; 
Conservation  and  Dissipation  of  Energy — Power,  23  ;  Problems  on 
Energy,  24. 

CHAPTER    III. 

COMPOSITION    AND    RESOLUTION    OF    MOTION. 

Motion  by  Two  or  More  Forces — The  Parallelogram  of  Forces,  25  ;  Ve- 
locities Represented — The  Triangle  of  Forces,  26  ;  The  Forces  Repre- 
sented Trigonometrically — Greatest  and  Least  Values  of  the  Resultant, 
27  ;  The  Polygon  of  Forces,  28  ;  Curvilinear  Motion — Calculation  of 
the  Resultant  of  Two  Impulsive  Forces,  29  ;  The  Resultant  and  all  the 


VI  CONTEXTS. 

Components,  except  one,  being  given,  to  Find  that  one  Component,  31  ; 
Resolution  of  Motion,  32  ;  Resolution  of  a  Force,  to  Find  its  Efficiency 
in  a  Given  Direction,  33  ;  Resultant  Found  by  Means  of  Rectangular 
Axes — Analytical  Expression  for  the  Resultant,  34  ;  Principle  of  Mo- 
ments, 35 ;  "Forces  Acting  at  Different  Points — Parallel  Forces,  37  ; 
Point  of  Application  of  the  Resultant,  38 ;  Equilibrium  of  Parallel 
Forces — Equilibrium  of  Couples — The  Parallelopiped  of  Forces,  39  ; 
Rectangular  Axes,  40 ;  Geometrical  Relation  of  Components  and  Re- 
sultant— Trigonometrical  Relation  of  Components  and  Resultant,  41  ; 
Any  Number  of  Forces  Reduced  to  Three  on  Three  Rectangular  Axes 
— Equilibrium  of  Forces  in  Different  Planes — Forces  Resisted  by  a 
Smooth  Surface,  42. 

CHAPTEK    IV. 

THE     CENTRE     OF     GRAVITY. 

The  Centre  of  Gravity  Defined — Centre  of  Gravity  of  Equal  Bodies  in  a 
Straight  Line,  43 ;  Centre  of  Gravity  of  Regular  Figures — Centre  of 
Gravity  between  Two  Unequal  Bodies,  44 ;  Equal  Moments  with  Respect 
to  the  Centre  of  Gravity — Centre  of  Gravity  between  Three  or  More 
Bodies — Centre  of  Gravity  of  a  Triangle — Centre  of  Gravity  of  an 
Irregular  Polygon,  45  ;  Centre  of  Gravity  of  the  Perimeter  of  an  Irreg- 
ular Polygon — Centre  of  Gravity  of  a  Pyramid,  46 ;  Examples  on  the 
Centre  of  Gravity,  47 ;  Centre  of  Gravity  of  Bodies  in  a  Straight  Line 
Referred  to  a  Point  in  that  Line — Centre  of  Gravity  of  a  System  Re- 
ferred to  a  Plane,  48  ;  Centre  of  Gravity  of  a  Trapezoid,  49  ;  Centrobaric 
Mensuration,  50  ;>  Examples,  52  ;  Support  of  a  Body — Different  Kinds  of 
Equilibrium,  53  ;  Stable  Equilibrium — Unstable  Equilibrium — Neutral 
Equilibrium,  54 ;  Questions  on  the  Centre  of  Gravity— Motion  of  the 
Centre  of  Gravity  of  a  System  when  One  of  the  Bodies  is  Moved,  55  ; 
Motion  of  the  Centre  of  Gravity  of  a  System  when  Several  of  the 
Bodies  are  Moved,  56  ;  Mutual  Action  among  the  Bodies  of  a  System — 
Examples  on  the  Motion  of  the  Centre  of  Gravity,  57. 

CHAPTEK    V. 

ELASTICITY. 

Elastic  and  Inelastic  Bodies— Mode  of  Experimenting— Collision  of  In- 
elastic Bodies,  59  ;  Questions  on  Inelastic  Bodies — Collision  of  Elastic 
Bodies,  60  ;  Questions  on  Elastic  Bodies,  61  ;  Impact  on  an  Immovable 
Plane— Imperfect  Elasticity,  62  ;  Elasticity  of  Traction— Elasticity  of 
Torsion,  63  ;  Elasticity  of  Flexure,  64. 

CHAPTEK    VI. 

SIMPLE     MACHINES. 

Classification  of  Machines,  65;  The  Three  Orders  of  Straight  Lever- 
Equal  Moments  in  Relation  to  the  Fulcrum,  66;  The  Acting  Distance, 
67  ;  Lever  not  Straight  and  Forces  not  Parallel — The  Compound  Lever, 
68:  The  Balance,  69;  The  Steelyard,  70;  Platform  Scales,  71  ;  Ques- 
tions on  the  Lever,  72  ;  Description  and  Law  of  the  Wheel  and  Axle, 
73  ;  Differential  Pulley,  74  ;  The  Compound  Wheel  and  Axle— Direc- 
tion and  Rate  of  Revolution.  75  ;  Questions  on  the  Wheel  and  Axle— 
The  Pulley  Described,  76  ;  The  Fixed  Pulley— The  Movable  Pulley— 
The  Compound  Pulley,  77  ;  Definition  and  Law  of  the  Rope  Machine, 
78;  Change  in  the  Ratio  of  Power  and  Weight,  79;  The  Branching 
Rope — Relation  of  Power,  Weight,  and  Pressure  on  the  Inclined  Plane, 
80  ;  Power  most  Efficient  when  Acting  Parallel  to  the  Plane— Expres- 


CONTENTS.  vji 

sion  for  Perpendicular  Pressure,  81  ;  Equilibrium  Between  Two  In- 
clined Planes — Bodies  Balanced  on  Two  Planes  by  a  Cord  Passing  over 
the  Ridge,  82  ;  Questions  on  the  Inclined  Plane — The  Screw  Reducible 
to  the  Inclined  Plane,  83 ;  The  Screw  and  Lever  Combined,  84 ;  The 
Endless  Screw— The  Right  and  Left  Hand  Screw,  85  ;  Questions  on  the 
Screw— Definition  of  the  Wedge,  and  the  Mode  of  Using— Law  of 
Equilibrium,  86  ;  Description  and  Law  of  Equilibrium  of  Knee  Joint, 
87  ;  Ratio  of  Power  and  Weight  Variable,  88  ;  Definition  of  Virtual  Ve- 
locity—The Point  of  Application  Moving  in  the  Line  of  the  Force,  89  ; 
The  Point  of  Application  Moving  in  a  Different  Line  from  that  in  which 
the  Force  Acts,  91  ;  The  Power  and  Weight  not  the  only  Forces  in  a 
Machine — Modes  of  Experimenting  on  Friction,  92  ;  Laws  of  Sliding 
Friction,  93  ;  Friction  of  Axes — Rolling  Friction,  94 ;  Advantages  of 
Friction — Limiting  Angle  of  Friction,  95. 

CHAPTER    VII. 

MOTION   ON   INCLINED   PLANES.— THE   PENDULUM. 

The  Force  which  Moves  a  Body  Down  an  Inclined  Plane,  96  ;  Formulae 
for  the  Inclined  Plane— Formulas  for  the  Whole  Length  of  a  Plane,  97  ; 
Descent  on  the  Chords  of  a  Circle,  98  ;  Velocity  Acquired  on  a  Series 
of  Planes — The  Loss  in  Passing  from  One  Plane  to  Another — No  Loss 
on  a  Curve,  99  ;  Times  of  Descending  Similar  Systems  of  Planes  and. 
Similar  Curves,  100 ;  Questions  on  the  Motions  of  Bodies  on  Inclined 
Planes— The  Pendulum,  101 ;  Calculation  of  the  Length  of  a  Pendu- 
lum, 102  ;  The  Point  of  Suspension  and  the  Centre  of  Oscillation  Inter- 
changeable, 103  ;  Calculation  of  the  Time  of  Oscillation,  104  ;  Applica- 
tions of  the  Formula,  106  ;  The  Compensation  Pendulum,  107. 

CHAPTER    VIII. 

CENTRAL     FORCES. 

Central  Forces  Described,  108;  Expressions  for  the  Centrifugal  Force  in 
Circular  Motion,  109;  Two  Bodies  Revolving  about  their  Centre  of 
Gravity— Centrifugal  Force  on  the  Earth's  Surface,  110;  Examples  on 
Central  Forces— Composition  of  Two  Rotary  Motions,  111 ;  The  Gyro- 
scope, 112. 


PART    II.— HYDRAULICS. 

CHAPTER    I. 

HYDROSTATICS. 

Liquids  Distinguished  from  Solids  and  Gases — Transmitted  Pressure,  114 ; 
—The  Hydraulic  Press,  115;  Equilibrium  of  a  Fluid,  116;  The  Curva- 
ture of  a  Liquid  Surface— The  Spirit  Level,  117  ;  Pressure  as  Depth,  118  ; 
Hydrostatic  Paradox,  119  ;  Amount  of  Pressure  in  Water,  120  ;  Column 
of  Water  whose  Weight  Equals  the  Pressure — Illustrations  of  Hydro- 
static Pressure,  121 ;  Determination  of  Thickness  of  Cylinder,  122  ;  The 
Same  Level  in  Connected  Vessels,  123;  Centre  of  Pressure — The  Loss 
of  Weight  in  Water,  124  ;  Equilibrium  of  Floating  Bodies.  125  ;  The 
Metaeenter,  126  :  Floating  in  a  Small  Quantity  of  Water— Floating  of 
Heavy  Substances— Specific  Gravity,  127  ;  Methods  of  Finding  Specific 
Gravity,  128;  The  Hydrometer.  or'Areometer,  129;  Specific  Gravity  of 
Liquids  by  Means  of 'Heights— Table  of  Specific  Gravities,  130;  Float- 


viii  CONTENTS. 

ins  131 ;  To  Find  the  Magnitude  of  an  Irregular  Body— Cohesion  and 
Adhesion— Capillary  Action,  132 ;  Capillary  Tubes— Parallel  and  In- 
clined Plates,  134 ;  Effects  of  Capillarity  on  Floating  Bodies— Illustra- 
tions of  Capillary  Action,  135  ;  Questions  in  Hydrostatics,  136. 

CHAPTER    II. 

HYDRODYNAMICS. 

Depth  and  Velocity  of  Discharge,  137;  Descent  of  Surface— Discharge 
from  Orifices  in  Different  Situations,  138  ;  Friction  in  Pipes— Jets,  140  ; 
Rivers,  141  ;  Hydraulic  Pumps,  142 ;  Centrifugal  Pumps— The  Hy- 
draulic Ram,  143 ;  Water- Wheels  with  a  Horizontal  Axis,  144 ;  The 
Turbine,  145  ;  Barker's  Mill— Resistance  to  Motion  in  a  Liquid,  147 ; 
Waves  — Molecular  Movements,  148;  Phases  — Wave  Length,  149; 
Water  Wave  Curve— Velocity  of  Propagation— Time  of  Oscillation-— 
Interference  of  Waves,  150  ;  Reflection  of  Waves — Sea  Waves,  151. 


PART    III.— PNEUMATICS. 

CHAPTEE    I. 

PROPERTIES  OF   GASES.— INSTRUMENTS   FOR   INVESTIGATION. 

Gases  Distinguished  from  Liquids— Tension  of  Gases,  152 ;  Change  of 
Condition  —  Diffusion  of  Gases  — Osmose  of  Gases,  153;  Weight  of 
(jases — Pressure  of  Gases— Buoyancy,  154;  Torricelli's  Experiment, 
155;  Pressure  of  Air  Measured— Pascal's  Experiment,  156;  Mariotte's 
Law,  157 ;  Dalton's  Law — Laws  of  Mixture  of  Gases  and  Liquids,  158  ; 
The  Barometer,  159  ;  Corrections  for  the  Barometer,  160  ;  The  Aneroid 
Barometer — Pressure  and  Latitude,  161  ;  Diurnal  Variation,  162 ;  The 
Barometer  and  the  Weather — Heights  Measured  by  the  Barometer — 
The  Gauge  of  the  Air-Pump,  163. 

CHAPTEE    II. 

INSTRUMENTS  WHOSE  OPERATION  DEPENDS  ON  THE  PROPERTIES 
OF  AIR. 

The  Air-Pump,  164  ;  Operation— Rate  of  Exhaustion,  165  ;  Sprengel's  Pump 
— The  Air  Condenser,  166  ;  Experiments  with  the  Air  Condenser— The 
Bellows,  167  ;  The  Siphon— Siphon  Fountain,  168  ;  The  Suction-Pump, 
169  ;  Calculation  of  Force,  170  ;  The  Forcing  Pump— The  Fire-Engine, 
171  ;  Hero's  Fountain,  172 ;  Manometers,  173 ;  Apparatus  for  Preserv- 
ing a  Constant  Level,  174  ;  Problems. 

CHAPTEE    III. 

THE    ATMOSPHERE.— ITS   HEIGHT    AND   MOTIONS. 

Virtual  Weight  of  the  Atmosphere,  175;  Decrease  of  Density— Actual 
Weight  of  the  Atmosphere,  176  ;  The  Motions  of  the  Air— The  Trade 
Winds,  177;  The  Return  Currents — Circulation  Beyond  the  Trade 
Winds,  178  ;  Land  and  Sea  Breezes— A  Current  Through  a  Medium- 
Ventilators,  179  ;  A  Stream  Meeting  a  Surface— Diminution  of  Pressure 
on  a  Surface,  180  ;  Vortices  where  the  Surface  Ends— Vortices  by  Cur- 
rents Meeting,  181. 


CONTENTS. 


PART    IV.  — ACOUSTICS. 

CHAPTEK    I. 

NATURE     AND     PROPAGATION     OP     SOUND. 

Sound— Vibrations,  182  ;  Sonorous  Bodies— Air  as  a  Medium  of  Sound — 
Method  of  Propagation  of  Sound,  183 ;  Huyghens's  Principle,  184 ; 
Velocity  of  Sound  in  Air — Newton's  Formula,  185  ;  Velocity  us  Affected 
by  the  Condition  of  the  Air  and  the  Quality  of  the  Sound,  186  ;  Other 
Gaseous  Bodies  as  Media  of  Sound — Liquids  as  Media,  187 ;  Solids  as 
Media,  188  ;  Velocity  in  Solids— Structure— Mixed  Media,  189 ;  Inten- 
sity of  Sound,  190 ;  Diffusion  of  Sound,  191. 

CHAPTEK    II. 

REFLECTION,    REFRACTION,    AND    INFLECTION    OF    SOUND. 

Reflection  of  Sound— Echoes,  192  ;  Simple  and  Complex  Echoes,  193  ;  Con- 
centrated Echoes — Resonance  of  Rooms,  194  ;  Halls  for  Public  Speak- 
ing, 195  ;  Refraction  of  Sound — Inflection  of  Sound,  196. 

CHAPTER    III. 

MUSICAL   SOUNDS    AND  MODES  OF   PRODUCING  THEM. 

Characteristics  of  Musical  Sounds — The  Pitch  of  Musical  Sounds — The 
Monochord  or  Sonometer,  197  ;  Time  of  a  Complete  Vibration,  198  ; 
The  Number  of  Vibrations  in  a  Given  Time— Vibration  of  a  String  in 
Parts,  199 ;  Longitudinal  Vibrations  of  Strings,  200 ;  Pitch  or  Fre- 
quency determined  by  Wave  Length— Resonant  Cavities,  201  ;  Station- 
ary Sound  Waves,  202 ;  Stopped  Organ  Pipes— Open  Organ  Pipes- 
Vibrations  of  a  Column  of  Air  in  Parts— Nodes,  203  ;  Modes  of  Exciting 
Vibrations  in  Pipes,  205;  Vibrations  of  Rods  and  Laminae — Wires, 
206;  Chladni's  Plates— Bells,  207;  The  Voice,  208;  The  Organ  of 
Hearing,  209. 

CHAPTEE    IV. 

MUSICAL    SCALES.— THE    RELATIONS    OF    MUSICAL    SOUNDS. 

Numerical  Relations  of  the  Notes,  210  ;  Relations  of  the  Intervals — Repe- 
tition of  the  Scale — Modes  of  Naming  the  Notes,  211  ;  The  Chromatic 
Scale,  212;  Chords  and  Discords — Temperament,  213;  Harmonics — 
Overtones — Timbre,  or  Quality  of  Tone.  215  ;  Communication  of  Vibra- 
tions, 216  ;  One  System  of  Vibrations  Controlling  Another — Crispations 
of  Fluids,  217;  Interference  of  Waves  of  Sound,  218;  Number  and 
Length  of  Waves  for  each  Note,  219  ;  Doppler's  Principle,  220  ;  Acous- 
tic Vibrations  Visibly  Projected— Edison's  Phonograph,  221. 

PART    V.  —  OPTICS. 

CHAPTER    I. 

MOTION    AND    INTENSITY    OF    LIGHT. 

Definitions— Light  Moves  in  Straight  Lines,  222 ;  The  Velocity  of  Light, 
223  ;  Determination  of  the  Velocity  of  Light  by  Experiment,  224  ;  Loss 
of  Intensity  by  Distance — Brightness  the  Same  at  nil  Distances,  225; 
Bunsen's  Photometer,  226  ;  Rumford's  Photometer — Shadows,  227. 


x  CONTENTS. 

CHAPTEK    II. 
REFLECTION     OF     LIGHT. 

Kadiant  and  Specular  Reflection,  228  ;  The  Law  of  Reflection,  229  ;  Inclina- 
tion of  Rays  to  Each  Other  not  Altered  by  the  Plane  Mirror,  230 ; 
Spherical  Mirrors — Converging  Effect  of  a  Concave  Mirror,  281  ;  Con- 
jugate Foci — Diverging  Effect  of  a  Convex  Mirror,  233 ;  Images  by 
Reflection — Images  by  a  Plane  Mirror,  235 ;  Symmetry  of  Object  and 
Image — The  Length  of  Mirror  Requisite  for  Seeing  an  Object — Displace- 
ment of  Image  by  Two  Reflections,  236 ;  Multiplied  Images  by  Two 
Mirrors,  237  ;  The  Kaleidoscope,  238  ;  Images  by  the  Concave  Mirror, 
239 ;  Illustrated  by  Experiment,  240 ;  Images  by  the  Convex  Mirror, 
241 ;  Caustics  by  Reflection— Spherical  Aberration  of  Mirrors,  243. 

CHAPTEK    III. 

REFRACTION    OF     LIGHT. 

Division  of  the  Incident  Beam— Refraction,  243 ;  Law  of  Refraction- 
Limit  of  Transmission  from  a  Denser  to  a  Rarer  Medium,  245  ;  Opacity 
of  Mixed  Transparent  Media — Transmission  through  Parallel  Plane 
Surfaces,  246 ;  Determination  of  Relative  Indices  of  Refraction  — 
Transmission  Through  a  Medium  Bounded  by  Inclined  Planes,  247 ; 
Prism  Used  for  Measuring  Refractive  Power — Light  through  One  Sur- 
face, 248  ;  Lenses,  250  ;  General  Effect  of  the  Convex  Lens,  251  ;  Gen- 
eral Effect  of  the  Concave  Lens— The  Optic  Centre  of  a  Lens,  252 ; 
Conjugate  Foci — To  Find  the  Principal  Focus,  253  ;  Powers  of  Lenses 
Practically  Determined,  254 ;  Equivalent  Combinations,  255  ;  Images 
by  the  Convex  Lens,  256  ;  Caustics  by  Refraction,  257  ;  Spherical  Aber- 
ration of  a  Lens — Remedy  for  Spherical  Aberration,  258  ;  Atmospheric 
Refraction— Mirage,  259. 

CHAPTER    IV. 
DECOMPOSITION   AND     DISPERSION     OF    LIGHT. 

The  Prismatic  Spectrum,  260  ;  The  Individual  Colors  of  the  Spectrum  can- 
not be  Decomposed  by  Refraction — Colors  of  the  Spectrum  Recom- 
bined,  261  ;  Complementary  Colors — Natural  Colors  of  Bodies— The 
Continuous  Spectrum,  262 ;  The  Line  Spectrum— Frauenhofer  Lines, 
263;  The  Spectroscope— Dispersion  of  Light,  264;  Chromatic  Aber- 
ration of  Lenses — Achromatism,  265  ;  Achromatic  Lens — Colors  not  Dis- 
persed Proportionally,  266. 

CHAPTER    V. 

RAINBOW     AND     HALO. 

The  Rainbow— Action  of  a  Transparent  Sphere  on  Light,  267  ;  The  Primary 
Bow— Course  of  Rays  in  Secondary  Bow,  268  ;  Axis  of  the  Bows— Cir- 
cular Form  of  the  Bows,  270 ;  Colors  of  the  Two  Bows  in  Reversed 
Order — Rainbows,  the  Colored  Borders  of  Illuminated  Segments  of  the 
Sky,  271  ;  The  Common  Halo— How  Caused,  272  ;  Its  Circular  Form— 
The  Halo,  a  Bright  Border  of  an  Illuminated  Zone — Frequency  of  the 
Halo— The  Mock  Sun,  273. 

CHAPTER    VI. 

NATURE     OF     LIGHT.— WAVE     THEORY. 

The  W.-ive  Theory— Nature  of  the  Wave,  274;  Postulates  of  the  Wave 
Theory — Reflection  and  Refraction  according  to  the  Wave  Theory,  275  ; 


CONTENTS. 


tion  of  Angles  of  Incidence,  Reflection,  and  Refraction,  276  ;  Inter- 
ice,  277  ;  Striated  Surfaces — Thin  Laminae,  278  ;  Newton's  Rings, 


Relation 
ference, 

279  ;  Thickness  of  Laminae  for  Newton's  Rings,  280  ;  Relation  of  Rings 
by  Reflection  aud  by  Transmission — Newton's  Rings  by  a  Monochro- 
matic Lamp — Diffraction  or  Inflection,  281  ;  Inflection  by  One  Edge  of 
an  Opaque  Body — Light  through  Small  Apertures,  282 ;  Why  Inflec- 
tion is  not  Always  Noticed  in  Looking  by  the  Edges  of  Bodies,  283 ; 
Length  and  Number  of  Luminous  Waves — Calorescence  and  Fluores- 
cence— Phosphorescence,  284. 

CHAPTEK    VII. 

DOUBLE     REFRACTION     AND     POLARIZATION. 

Change  of  Vibrations  in  Polarized  Light — Polarizing  and  Analyzing  by  Re- 
flection— Polarization  by  Reflection,  285 ;  Changes  of  Intensity  'De- 
scribed, 286 ;  The  Polarizing  Angle— Polarization  of  a  Bundle  of 
Plates— Polarization  by  Absorption,  287  ;  Double  Refraction— Ordinary 
and  Extraordinary  Rays,  288  ;  Polarizing  by  Double  Refraction — Dif- 
ferent Kinds  of  Polarization,  289 ;  Every  Polarizer  an  Analyzer — 
Nicol's  Prism,  290 ;  Color  by  Polarized  Light,  291  ;  Rotation  of  the 
Plane  of  Polarization,  293. 

CHAPTER    VIII. 
VISION. 

Image  by  Light  Through  an  Aperture — Effect  of  a  Convex  Lens  at  the  Aper- 
ture, 294 ;  The  Eye— The  Interior  of  the  Eye,  295  ;  Vision— Adapta- 
tions, 296  ;  Accommodation  to  Diminished  Distance,  297  ;  Long-sighted- 
ness— Short-sightedness,  298  ;  Why  an  Object  is  seen  Erect  and  Single 
— Indirect  Vision — The  Blind  Point,  299  ;  Continuance  of  Impressions 
— Subjective  Colors,  300;  Irradiation — Estimate  of  the  Distance  of 
Bodies— Magnitude  and  Distance  Associated.  301  :  Binocular  Vision— 
The  Stereoscope,  302. 

CHAPTER    IX. 

OPTICAL     INSTRUMENTS. 

The  Camera  Lucida,  302 ;  The  Microscope— The  Single  Microscope,  303 ; 
The  Compound  Microscope — The  Magnifying  Power — Modern  Im- 
provements, 305  ;  The  Telescope — The  Astronomical  Telescope,  307  ; 
The  Powers  of  the  Telescope,  308 ;  The  Terrestrial  Telescope,  309  ; 
Galileo's  Telescope — The  Gregorian  Telescope,  310 ;  The  Newtonian 
Telescope— The  Herschelian  Telescope,  311 ;  Eye-pieces,  or  Oculars,  312. 


PART    VI.  — HEAT. 

CHAPTER    I. 

EXPANSION    BY    HEAT.— THE    THERMOMETER. 

Nature  of  Heat— Expansion  and  Contraction  by  Heat  and  Cold — Expansion 
of  Solids,  313  ;  Coefficient  of  Expansion — The  Coefficient  of  Expansion 
differs  in  Different  Substances,  314  ;  The  Strength  of  the  Thermal 
Force— Expansion  of  Liquids,  315  ;  Exceptional  Case — Expansion  of 
Gases,  316;  The  Thermometer,  317;  Different  Systems  of  Graduation 
—To  Reduce  from  One  Scale  to  Another— Absolute  Zero  of  Tempera- 
ture, 318. 


xii  CONTENTS. 

CHAPTEK    II. 

PASSAGE    OF    HEAT    THROUGH    MATTER    AND    SPACE. 

Heat  is  Communicated  in  Several  Ways — Conduction  of  Heat  by  Solids, 
319  ;  Effects  of  Molecular  Arrangement,  320  ;  Conduction  by  Fluids — 
Illustrations  of  Difference  in  Conductive  Power — Convection  of  Heat. 
321  ;  Determination  of  the  Temperature  of  Water  at  its  Maximum  Den- 
sity, 322 ;  Radiation  of  Heat,  323  ;  Equalization  of  Temperature— Re- 
flection of  Heat — Heat  Concentrated  by  Reflection,  324  ;  Absorption  of 
Heat — Diathermancy,  325. 

CHAPTEE    III. 

SPECIFIC    HEAT.— CHANGES    OF    CONDITION.— LATENT    HEAT. 

Specific  Heat,  326 ;  Method  of  Finding  Specific  Heat,  327 ;  Apparent  Con- 
duction Affected  by  Specific  Heat  —  Changes  of  Condition  — Latent 
Heat,  328 ;  Fusion  or  Melting,  329  ;  Vaporization— Other  Causes  Af- 
fecting the  Boiling  Point,  330  ;  Spheroidal  Condition — Evaporation — 
Condensation,  331  ;  Solidification— Freezing  Produced  by  Melting,  332  ; 
Freezing  by  Evaporation — Regelation,  333. 

CHAPTEE    IV. 

TENSION  OF  VAPOR.— THE  STEAM  ENGINE.— MECHANICAL  EQUIV- 
ALENT   OF    HEAT. 

Dalton's  Laws  —  Experimental  Illustration,  334;  Tensions  of  Different 
Vapors — Tension  in  Generator  and  Condenser,  335  ;  Heat  Energy  in 
Steam— Tension  of  Steam— The  Steam  Engines  of  Savery  and  New- 
comen,  336  ;  The  Steam  Engine  of  Watt— The  Double-acting  Engine, 
338  ;  Condensing  Engine,  339  ;  Non-condensing  Engine — Calculation  of 
Steam  Power,  340 ;  Mechanical  Equivalent  of  Heat,  841. 

CHAPTEE    V. 

TEMPERATURE    OF    THE    ATMOSPHERE.— MOISTURE    OF    THE    AT- 
MOSPHERE.—DRAFT    AND    VENTILATION. 

Manner  in  which  the  Air  is  Warmed — Limit  of  Perpetual  Frost — Isothermal 
Lines,  342  ;  Moisture  of  the  Atmosphere — Temperature  and  Tension  of 
Vapor,  343  ;  Dew  Point — Measure  of  Vapor — Hygrometers — Dew — 
Frost,  344  ;  Fog,  345 ;  Cloud— Rain— Mist,  346  ;  Hail,  Sleet,  Snow- 
Draught  of  Flues,  347 ;  Ventilation  of  Apartments,  348 ;  Sources  of 
Heat,"349. 


PART    YIL— ELECTRICITY    AND    MAG- 
NETISM. 

CHAPTEE    I. 

ELECTROSTATICS. —POTENTIAL.— CAPACITY. 

Definition — Common  Indications  of  Electricity — Repulsion,  351  ;  Theories 
of  Electricity — Electric  Series,  352  ;  Conductors  and  Insulators — Cou- 
lomb's Law,  353 ;  Potential.  354 ;  Equipotential  Surfaces,  355 ;  Dif- 
ference of  Potential— Unit  of  Potential,  356  ;  Zero  Potential— Potential 
on  a  Sphere — Capacity,  357  ;  Equipotential  of  Connected  Conductors — 
Position  of  Static  Charge — Distribution  of  a  Chnrge  on  the  Surface, 
359;  Surface  Density— Quadrant 'Electrometer,  360;  Problems,  361. 


CONTENTS.  xiii 

CHAPTER    II. 

ELECTROSTATIC     INDUCTION. 

Gold-leaf  Electroscope — Phenomena,  862  ;  Induction  Precedes  Attraction, 
363  ;  Quantity  of  the  Induced  Electricity— Condensers,  364 ;  Specific 
Inductive  Capacity,  365  ;  Leyden  Jar— Seat  of  the  Charge,  367  ;  Resid- 
ual Charge — Modern  Theory  of  Condensers,  368  ;  Hertz's  Experiments, 
369  ;  Electrical  Machines— Electrophorus— Holtz  Machine,  370  ;  Effects 
of  Statical  Discharge— Lightning,  372. 

CHAPTER    III. 

MAGNETISM. 

Natural  Magnets — Artificial  Magnets — Poles  of  a  Magnet — Magnetic  Needle, 
373  ;  Attractions  and  Repulsions — North  and  South  Poles  Inseparable — 
Magnetic  Induction,  374 ;  Retentivity  or  Coercive  Force,  375  ;  Law  of 
Magnetic  Force — Unit  Magnet  Pole,  376  ;  Lifting  Power — Laminated 
Magnets,  377 ;  Magnetic  Field— Lines  of  Force,  378 ;  Theory  of  the 
Curvature  of  the  Lines,  379;  Fields  from  Several  Magnets,  380; 
Strength  or  Intensity  of  Magnetic  Field — Determination  of  the  Strength 
of  a  Field,  381  ;  Hysteresis — Number  of  Lines  of  Force  from  a  Given 
Pole,  382  ;  Magnetic  Susceptibility,  383  ;  Magnetic  Permeability— Mag- 
netic Circuit,  384 ;  Paramagnetism  and  Diamaguetism,  385. 

CHAPTER    IV. 

TERRESTRIAL     MAGNETISM. 

The  Earth  a  Magnet— Declination  of  the  Needle— Isogonic  Curves,  386; 
Secular  and  Annual  Variation  —  Diurnal  Variation  —  Magnetometer, 
388  ;  Dip  of  the  Needle,  389  ;  Isoclinic  Curves,  390  ;  Intensity  of  the 
Earth's  Magnetism,  391;  Isodynamic  Curves,  392;  Variation  in  the 
Strength  of  the  Earth's  Field— Astatic  Needles,  393  ;  Magnetic  Charts 
— The  Declination  Compass — The  Mariner's  Compass,  394 ;  Aurora 
Borealis— Why  the  Earth  is  a  Magnet,  395. 

CHAPTER    V. 
CURRENT     ELECTRICITY. 

Electricity  in  Motion— Galvanic  Cells.  396;  Electromotive  Force,  397; 
Polarization— Types  of  Batteries,  398  ;  Daniell's  Cell— Leclanche  Cell, 
399  ;  Combustion  of  Zinc — Amalgamation  of  Zincs,  400  ;  Practical 
Units  of  Current  and  Quantity — Resistance,  401 ;  Influence  of  Tem- 
perature, 402  ;  Ohm's  Law — Divided  Circuits — Shunts,  403  ;  Ratio  of 
Currents  in  Shunts— Fall  of  Potential,  404  ;  Resistance  Boxes  or  Rheo- 
stats— Wheatstone's  Bridge,  405  ;  Cells  in  Series  and  in  Multiple  Arc, 
407  ;  Problems,  408. 

CHAPTER    VI. 

ELECTRO-MAGNETISM. 

The  Current's  Lines  of  Magnetic  Force,  409  ;  Effect  of  a  Current  on  a  Mag- 
net, 410  ;  Solenoids,  411 ;  Ampere's  Theory  of  Magnetism,  412  ;  Electro- 
mairnets-Magneto-motive  Force,  413  ;  The  Morse  Tclogr.-iph  System, 
415;  The  Relay,  417;  Duplex  Telegraphy,  4LS;  Atlantic  Telegraph 
Cable— Electric'Bells,  420  ;  Galvanometers,  421. 


xiv  CONTENTS. 


CHAPTEK    VII. 

EL  EOT  BO -DYNAMICS. 

Movements  of  Conductors  Carrying  Currents — Parallel  Currents,  423  ;  Cur- 
rents not  Parallel,  424  ;  Continuous  Rotation  Produced  by  Mutual  Ac- 
tion of  Currents,  425  ;  Electro-dynamometer,  426. 


CHAPTEK    VIII. 

ELECTRO-MAGNETIC   INDUCTION. 

Currents  of  Electricity  Produced  by  Induction — Methods  of  Producing  the 
Inducing  Field,  427  ;  Lenz's  Law— Self-induction,  429  ;  Coefficients  of 
Mutual  and  Self-induction— Induced  Currents  from  the  Earth,  430  ; 
Arago's  Rotations— Induction  Colls,  431  ;  The  Telephone,  432 ;  The 
Blake  Transmitter,  433  ;  Dynamos,  434  ;  Electric  Motors,  436. 


CHAPTEE    IX. 

ELECTRO-CHEMISTRY    AND    ELECTRO-OPTICS. 

Electrolytes,  436 ;  Electrolysis  of  Sulphuric  Acid— Metallic  Salts,  437 ; 
Faraday's  Laws — Voltameters,  438  ;  Theory  of  Electrolysis — Electro- 
plating, 439;  Electrotyping —  Counter  -  Electromotive  Force,  440; 
Storage  Batteries,  441  ;  Capillary  Electrometer,  442  ;  Light  and  Elec- 
tricity— Double  Refraction  from  Electrostatic  Strain,  443 ;  Magneto- 
optic  Twisting  of  the  Plane  of  Polarized  Light — Rotation  of  the  Plane 
by  Reflection — Photo-Electric  Properties  of  Selenium,  444. 


CHAPTEK    X. 

THE  RELATIONS  BETWEEN  ELECTRICITY  AND  HEAT. 

Power  of  the  Electrical  Current,  445  ;  Heat  Developed  in  a  Conductor — 
Rise  in  Temperature  of  the  Conductor,  446 ;  Hot  Wire  Ammeters  and 
Voltmeters,  447  ;  Electric  Welding— The  Electric  Arc,  448 ;  Incan- 
descent Electric  Lamps  —  Thermo-Electricity,  449  ;  Thermo-Electric 
Pile— Peltier  Effect,  450  ;  The  Electrical  Units,  451. 


APPENDIX. 

APPLICATIONS    OF    THE    CALCULUS. 

Falling  Bodies.  453 ;  Centre  of  Gravity,  456 ;  Centre  of  Oscillation,  460 
Centre  of  Hydrostatic  Pressure,  462  ;  Angular  Radius  of  the  Rainbow 
and  the  Halo,  464. 


NATURAL   PHILOSOPHY. 


INTRODUCTION. 

Art.  1.  Definition. — Physics,  or  Natural  Philosophy,  is  the 
science  of  matter  and  energy.  The  broadness  of  this  definition 
makes  the  science  include  three  special  departments,  which,  .fi-om. 
the  great  number  of  closely  related  facts  belonging  to  them,  are 
considered  as  distinct  sciences.  They  are  Biology,  which  treats 
of  matter  endowed  with  the  principle  of  life  ;  Chemistry,  treating 
of  the  inner  mechanism  of  ultimately  divided  matter  ;  and  Astron- 
omy, treating  of  gross  matter  in  the  form  of  worlds. 

Extensive  developments  have  rendered  it  necessary  to  limit  the 
domain  of  Physics,  until  it  is  now  popularly  considered  as  treating 
of  those  things  which  pertain  to  Mechanics,  Sound,  Heat,  Light, 
Electricity,  and  Magnetism. 

2.  Definitions  Relating  to  Matter. — 

A  Body  is  a  separate  portion  of  matter,  whether  large  or  small. 

An  Atom  is  a  portion  of  matter  so  small  as  to  be  indivisible. 

A  Particle  denotes  the  smallest  portion  which  can  result  from 
division  by  mechanical  means,  and  consists  of  many  atoms  united 
together. 

A  Molecule  is  the  smallest  portion  of  any  substance  which  can 
exist  in  a  free  state,  and  is  made  up  of  atoms. 

Mass  is  the  quantity  of  matter  iu  a  body,  and  is  usually  meas- 
ured by  its  weight. 

Volume  signifies  the  space  occupied  by  a  body. 

Density  expresses  the  relative  mass  contained  within  a  given 
volume.  Thus,  if  one  body  has  twice  as  great  a  mass  within  a  cer- 
tain volume  as  another  has,  it  is  said  to  have  twice  the  density. 

Pores  are  the  minute  portions  of  space  within  the  volume  of  a 
body  which  are  not  filled  by  the  material  of  that  body.  All  matter 
is  porous,  some  kinds  in  a  greater  and  some  in  a  less  degree. 


2  MECHANICS. 

3.  Properties  of  Matter. — 

(1.)  Extension. — Every  portion  of  matter,  however  small,  has 
length,  breadth,  and  thickness,  and  thus  occupies  space.  This  is 
its  extension. 

(2.)  Impenetrability. — While  matter  occupies  space  it  excludes 
all  other  matter  from  it,  so  that  no  two  atoms  can  be  in  exactly  the 
same  place  at  the  same  time.  This  property  is  called  impenetra- 
bility. 

The  two  foregoing  are  often  called  essential  properties,  because 
we  cannot  conceive  matter  to  exist  without  them. 

(3.)  Divisibility. — Matter  is  divisible  beyond  any  known  limits. 
After  being  divided,  as  far  as  possible,  into  particles  by  mechanical 
methods,  it  may  be  still  further  reduced  by  chemical  action  to 
atoms,  which  are  too  small  to  be  in  any  way  recognized  by  the 


(4.)  Compressibility. — Since  pores  exist  in  all  matter,  it  may  be 
compressed  into  a  smaller  volume.  Hence  all  matter  is  compressi- 
ble, though  in  very  different  degrees. 

(5.)  Elasticity. — After  a  body  has  suffered  compression,  it  shows, 
in  some  degree  at  least,  a  tendency  to  restore  itself  to  its  former 
volume.  This  property  is  called  elasticity.  A  body  is  said  to  be 
perfectly  elastic  when  the  force  by  which  it  recovers  its  size  is  equal 
to  that  by  which  it  was  before  compressed.  The  word  elasticity  is 
used  generally  in  a  wider  sense  than  is  given  in  the  above  definition, 
namely,  the  tendency  which  a  body  has  to  recover  its  original  form, 
whatever  change  of  form  it  may  have  previously  received.  Thus, 
if  a  body  is  stretched,  bent,  twisted,  or  distorted  in  any  other  way, 
it  is  called  elastic,  if  it  tends  to  resume  its  form  as  soon  as  the  force 
which  altered  it  has  ceased.  Torsion  is  the  name  of  the  elastic 
force  which  tends  to  untwist  a  thread  or  wire  when  it  has  been 
twisted. 

(6.-)  Attraction. — This  is  the  general  name  used  to  express  the 
universal  tendency  of  one  portion  of  matter  toward  another.  It 
receives  different  names,  according  to  the  circumstances  in  which 
it  acts.  The  attraction  which  binds  together  atoms  of  different 
kinds,  so  as  to  form  a  new  substance,  is  called  affinity,  and  is  dis- 
cussed in  Chemistry  ;  that  which  unites  particles,  whether  simple 
or  compound,  so  as  to  form  a  body,  is  called  cohesion  ;  the  clinging 
of  two  kinds  of  matter  to  each  other,  without  forming  a  new  sub- 
stance, is. called  adhesion;  and  the  tendency  manifested  by  masses 
of  matter  toward  each  other,  when  at  sensible  distances,  is  called 
gravitation. 

(7.)  Inertia. — This  is  also  a  universal  property  of  matter,  and 
signifies  its  tendency  to  continue  in  its  present  condition  as  to 


INTRODUCTION.  3 

motion  or  rest.  If  at  rest,  it  cannot  move  itself  ;  if  in  motion,  it 
•cannot  stop  itself  or  change  its  motion,  either  in  respect  to  direc- 
tion or  velocity. 

4.  The  Absolute  Units. — Physics  is  essentially  a  science 
of  measurement.  Everything  treated  in  it  is  the  direct  result  of 
observation.  All  its  laws  have  been  derived  from  patient  measure- 
ment. 

If  all  lengths,  from  the  diameter  of  an  atom  to  the  distance 
between  us  ami  the  farthest  star,  could  be  directly  measured  ;  if  all 
masses,  from  the  weight  of  an  atom  to  the  weight  of  the  sun,  could 
be  accurately  determined,  and  if  every  conceivable  duration  of  time 
could  be  measured,  then  all  physical  measurements  would  demand 
but  three  units,  viz.,  a  unit  of  length,  a  unit  of  mass,  and  a  unit  of 
time.  This  peculiar  property  of  these  magnitudes  earns  for  their 
respective  units  the  name  Absolute  Units. 

The  absolute  unit  of  length  is  the  centimeter  ;  of  mass,  the  gram; 
of  time,  the  second. 

Other  units,  whose  determination  involves  the*  use  cf  more  than 
one  absolute  unit,  are  called  derived  units.  Thus,  velocity  requires 
that  length  and  time  be  determined. 

Temperature,  which  we  now  measure  by  thermometers,  would, 
in  all  probability,  be  measured  as  a  length,  in  absolute  units.  This 
length  would  be  the  distance  through  which  a  molecule  vibrates. 


PART   I. 

MECHANICS 


CHAPTER    I. 

MOTION     AND     FORCE. 

5.  Classification  of  Motions. — Motion  is  change  of  place, 
and  may  be  classified  as  follows  : 

Uniform  Motion,  when  equal  spaces  are  passed  over  in  equal 
times ; 

Accelerated  Motion,  when  the  spaces  described  in  equal  times 
become  continually  greater  ; 

Retarded  Motion,  when  the  space's  described  in  equal  times  be- 
come continually  less. 

In  the  last  two  cases,  if  the  increments  or  decrements  of  space 
for  equal  times  are  equal,  the  motion  is  uniformly  accelerated  or 
retai-ded.  / 

The  space  described  by  a  body  moving  uniformly  in  the  unit 
of  time  is  termed  the  Velocity.  Accordingly  the  unit  velocity  would 
be  when  a  unit's  length  was  passed  in  a  unit  time,  e.g.,  centimeter 
per  second  or  foot  per  second. 

6.  Uniform  Motion. — When  motion  is  uniform,  the  numbei* 
of  centimeters  described  in  one  second,  multiplied  by  the  number 
of  seconds,  obviously  gives  the  whole  space.     Let  s  =  space,  t  = 
time,  and  v  =  velocity  ;  then  s  =  t  v  ;  whence 


7.  Questions  on  Uniform  Motion. — 

1.  A  ball  was  rolled  on  the  ice  with  a  velocity  of  780  centime- 
ters per  second,  and  moved  uniformly  21  seconds  ;  what  space  did 
it  describe  ?  .4ns.  16,380  cms. 

2.  A  steamboat  moved  uniformly  across  a  lake  17  miles  wide  at 


MOMENTUM.  5 

the  rate  of  20  feet  per  second  ;  what  time  was  occupied  ill  cross- 
ing? Am.  Ih.  14m.  48s. 

3.  On  the  supposition  that  the  earth  describes  an    orbit  of 
600,000,000  of  miles  in  365£  days,  with  what  velocity  does  it  move 
per  second?  An*.  19  miles,  nearly. 

4.  Three  planets  describe  orbits  which  are  to  each  other  as  15, 
19,  and  12,  in  times  which  are  as  7,  3,  and  5  ;  what  are  their  rdn- 
tice  velocities  ?  Ans.  225,  665,  and  252. 

8.  Momentum. — The  product  of  the  mass  of  a  body  and  its 
velocity  is  called  Momentum.     Thus  let  k  =  momentum,  m  —  the 

mass,    and   v  =    the    velocity,  and   we   have  k  =  m  v,  m  =  —,  and 

c 


If  the  momentum  of  one  body  equals  that  of  another,  then, 
since  k  =  k  ',  m  v  —  m'  v',  .'.  m  :  m'  :  :  v'  :  v.  That  is,  in  order  that 
the  momenta  of  two  bodies  should  be  equal,  their  masses  must 
vary  inversely  as  their  velocities. 

9.  Questions  on  Momentum.  — 

1.  A  cannon-ball  weighing  12  kilos.,  with  a  velocity  of  820  meters 
per  second,  hits  a  ship  with  what  momentum  ? 

2.  A  bill  weighing  10  grim.s  is  tired  into  a  log  weighing  9,990 
grams,  suspended  so  as  to  move  freely,  and  imparts  a  velocity  of  1 
meter  per  second.     Assuming  that  the  log  and  ball  have  a  momen- 
tum equal  to  the  previous  momentum  of  the  ball  alone,  required 
the  velocity  of  the  ball?  Ans.  1  kilometer  per  second. 

3.  Suppose  a  comet,  whose  velocity  is  1,X)00,000  miles  per  hour, 
has  the  same  momentum  as   the  earth,  whose  velocity  is  19  miles 
per  second  ;  what  is  the  ratio  of  their  masses?          Ans.  1  :  14.6. 

4.  Two  railway  cars  have  their  quantities  of  matter  as  7  to  3, 
and  their  momenta  as  8  to  5  ;  what  are  their  relative  velocities? 

Ans.  As  24  to  35,  or  nearly  5  to  7. 


10.  Force.  —  Force  w  tJtat  which  tends  to  pra/ucv,  alter,  or  de- 
stroy motion. 

Gravity,  friction,  explosions,  elasticity,  and  magnetic  attractions 
are  forces. 

We  say  the  tendency  of  a  force  is  to  produce  motion.  It  does 
not  always  do  so,  for  the  body  acted  upon  may  be  rigidly  restrained 
from  moving  by  an  equal  opposite  force,  which  comes  into  existence 
only  as  the  first  force  commences  to  act.  Thus  one  may  exert  a 
force  to  pull  a  nail  out  of  a  hard-wood  plank.  The  instant  the 


6  MECHANICS. 

force  of  the  pull  is  exerted  an  opposite  equal  force  of  friction  is 
generated. 

According  to  the  duration  of  time  which  a  force  acts  it  is  classed 
as 

An  Impulsive  force,  when  acting  for  an  inappreciable  length  of 
time,  or 

A  Continued  force,  when  acting  for  a  sensible  length  of  time. 

The  blows  of  a  hammer,  explosions,  and  electric  discharges  are 
impulsive  forces,  while  gravitation,  magnetic  attractions,  and  winds 
are  continued  forces. 

•When  the  strength  of  a  force  remains  unaltered  with  the  time 
it  is  called  a  Constant  force.  Such  a  one  is  gravity. 

11.  Motions  Produced  by  Force. — If  a  free  body  at  rest  be 
acted  upon  by  a  single  force  we  find  that 

An  impulsive  force  causes  uniform  motion,  and 
A  continued  force  causes  accelerated  motion.     If  the  continued 
force  be  constant,  the  resulting  motion  is  uniformly  accelerated. 

Of  course  it  is  impossible  to  practically  realize  the  motion  re- 
sulting from  a  single  force.  A  ball  sent  by  an  impulsive  discharge 
is  subject  not  only  to  the  influence  of  gravity,  but  also  to  the  force 
of  friction  from  the  air.  Even  gravity  has  to  work  against  the  force 
of  friction. 

12.  Measure  of  Force. — In   order   to   measure  such  an  in- 
tangible thing  as  force,  we  must  look  to  the  effects   which  it  has, 
and  measure  them. 

IMPULSIVE  FORCES. — In  the  case  of  an  impulsive  force  this  is 
easy,  for  it  gives  to  the  object  acted  upon  a  uniform  velocity.  Both 
the  body  and  the  velocity  must  be  considered.  Experiments  with 
the  same  force,  but  different  bodies,  show  that  the  greater  the  mass 
of  the  body  the  less  the  velocity  which  will  be  imparted  to  it. 
Furthermore,  it  will  be  found  that,  with  the  same  force,  the  prod- 
uct of  mass  and  velocity  will  be  constant.  Whence,  representing 
force  by  F,  we  have 

F  cc  m  u, 

But  (Art.  8)  k  x  m  v. 

Hence  an  impulsive  force  is  measured  by  the  momentum  it  produces 
or  destroys. 

CONSTANT  FORCES. — The  effects  of  a  constant  force  are  more 
numerous.  A  constant  force  like  the  pull  of  a  magnet  (through 
short  distances)  can  be  made  to  extend  an  elastic  spring  by  a 
measurable  amount  and  maintain  the  extension.  It  can  be  made 


MOTION    AXD    FORCE.  7 

to  oppose  the  constant  force  of  gravity,  as  by  attracting  one  end  of 
a  balance.  It  can  be  made  to  impart  a  uniformly  accelerated  mo- 
tion to  a  given  mass.*  All  these  methods  are  practically  employed 
in  the  measurement  of  constant  fqi'ces.  The  elasticity  of  a  spring 
is,  however,  too  complicated  to  be  used  in  obtaining  a  unit  of  force. 
Gravity  differs  (Art.  16)  at  various  parts  of  the  earth.  Hence  use 
is  made  of  the  accelerated  motion  imparted  to  a  given  mass. 

Of  two  different  constant^  forces,  working  for  the  same  length 
of  time,  the  stronger  will  give  a  greater  acceleration  to  the  same 
mass,  and  will  give  the  same  acceleration  to  a  greater  mass.     Let- 
ting a  =•  acceleration,  this  is  expressed  by 
F  oc   in  a. 

To  get  our  unit  of  force  we  change  to  -an  equality, 
F  =.  m  a. 

Making  m  and  a  units  we  have  for  the  unit  force  one  that  will 
produce  a  unit  acceleration  on  a  unit  mass.  Unless  we  know  what 
a  unit  acceleration  is,  we  are  still  at  sea. 

Acceleration  is  gain  of  velocity,  i.e.,  difference  between  the  ve- 
locity at  the  beginning  of  a  period  of  time  and  the  velocity  at 
the  end  of  that  period.  The  longer  the  period  the  greater  the 
gain,  hence  a  unit  period  (second)  should  be  taken.  Further,  it  is 
well  to  consider  the  first  velocity  as  zero,  i.e.,  the  body  to  be  at 
rest  when  first  acted  upon.  Thus  a  unit  acceleration  is  a  unit  ve- 
locity gained  in  a  unit  time.  As  an  equality, 


Substituting  in  the  equation  above  we  have 
_  mv 

Z    t   ' 

Expressing  all  these  quantities  in  absolute  units,  we  have  the  ab- 
solute unit  of  force. 

The  Dyne  is  that  force  which,  acting  upon  one  gram  fur  o)>e  sec- 
ond, will  produce  a  velocity  of  one  centimeter  per  second. 

To  measure  an  impulsive  force  in  this  manner  would  necessitate 
the  measurement  of  the  infinitely  short  time  that  it  acts.  It  really 
produces  an  accelerated  motion  while  it  acts,  and  the  uniform 
velocity  which  we  observe  is  the  gain  during  that  short  time. 

(The  force  exerted  by  the  earth  upon  one  pound  of  matter  is 
oftentimes  taken  as  the  unit  of  force.) 


*  On  the  measurement  of  constant  forces  by  the  method  of  oscillations, 
see  Art.  163. 


8  MECHANICS. 

QUESTIONS  ON  FORCE.  - 

1.  How  many  dynes  are  required  to  set  a  mass  weighing  50 
kilos,  in  motion  with  a  velocity  of  12  meters  per  second,  the  force 
being  supposed  to  act  for  precisely  one  second?     Ans.  60,000,000. 

2.  How  many  dynes  are  required  to  move  a  gram  9.81  meters 
per  second,  the  force  acting  for  1  second  ?     Acting  for  2  seconds  ? 

Ans.  981;  490.5. 

3.  Ten  dynes  act,  for  a  second,  on  a  kilogram.     What   is  its 
velocity  ? 

13.  The  Three  Laws  of  Motion.— All  the  phenomena  of 
motion  in  Mechanics  and  Astronomy  are  found  to  be  in  accordance 
with  three  first  principles,  which  Newton  announced  in  his  Princi- 
pia,  and  which  are  to  be  regarded  as  forming  the  basis  of  mechan- 
ical science.  They  may  be  named  and  defined  as  follows  : 

1.  The  law  of  Inertia. — A  body  at  rest  tends  to  remain  at  rest ; 
and  a  body  in  motion  tends  to  move  forever,  in  a  straight  line,  and 
uniformly. 

That  a  book  allows  a  paper  to  be  withdrawn  from  under  it, 
without  itself  moving,  is  because  of  its  inertia.  The  planets  con- 
tinue their  motion  because  of  their  inertia. 

2.  The  law  of  the   Coexistence  of  Motions. — A  body  subjected 
to  several  motions  will  ultimately  be  in  the  same  place,  whether 
these  motions  take  place  at  the  same  time  or  successively. 

A  boat,  under  the  influence  of  wind  and  tide,  would  be  in  the 
same  place  at  the  end  of  an  hour,  as  if  the  wind  acted  alone  for 
an  hour  and  the  tide  alone  for  another  hour.  This  law  is  fully 
discussed  in  Chapter  III. 

3.  The  law  of  Action  and  Reaction. — If  any  kind  of  action  takes 
place  between  two  bodies,  it  produces  equal  momenta  in  opposite 
directions  ;  or,  every  action  is  accompanied  by  an  equal  and  oppo- 
site .reaction. 

This  law  is  illustrated  by  the  kick  of  a  gun.  If  the  gun  were 
suspended  by  long  wires  and  then  fired,  it  would  be  found  that 
the  momentum  of  the  bullet  would  exactly  equal  the  momentum  of 
the  gun.  Owing  to  the  difference  between  the  weight  of  the  bullet 
and  of  the  gun,  the  velocity  of  the  former  is  much  the  greater. 
In  the  collision  of  two  railroad  trains,  it  is  immaterial  as  to  the 
effects  which  they  will  respectively  suffer,  Avhether  each  is  moving 
towards  the  other,  or  whether  one  is  at  rest,  provided  that  in  the 
latter  case  the  moving  train  has  a  momentum  equal  to  the  mo- 
menta of  the  two  trains  in  the  former  case.  When  a  magnet 
attracts  a  piece  of  iron,  each  moves  towards  the  other  with  the 
same  momentum. 


FOECE    OF    GRAVITY.  9 

14.  Force  of  Gravity. — Every  mass  of  matter  near  the  earth, 
-when  free  to  move,  pursues  a  straight  line  towards  its  centre.    The 
force  by  which  this  motion  is  produced  is  called  gravity  ;  either  the 
gravity  of  the  body  or  the  gravity  of  the  earth  ;  for  the  attraction 
is  mutual  and  equal,  in  accordance  with  the  third  law  of  motion. 
It   is   easy    to   understand  why   a   small    mass    should    attract  a 
large  one,  as  much  as  the  large  mass  attracts  the  small  one.     Let 
A  consist  of  one  atom  of  matter,  and  /?,  at  any  distance  from   it, 
consist  of  ten  atoms.     If  it  be  admitted  that  A  attracts  one  atom  of 
B  as  much  as  that  one  atom  attracts  A,  then  the  above  conclusion 
follows.     For  A  attracts  each  of  the  ten  atoms  of  B  as  much  as 
each  of  the  same  ten   attracts  A ;  so  that  A   exerts  ten  units  of 
attraction  dn  B,  while  B  exerts  ten  units  of  attraction  on  A.     The 
same  reasoning  obviously  applies  to   the  earth  in  relation  to  the 
small  bodies  on  its  surface. 

15.  Relation  of  Gravity  and  Mass. — At  the  same  distance 
from  the  centre  of  the  earth,  gravity  varies  as  the  mass.     This  is 
because  it  operates  equally  on  every  atom  of  a  body  ;  hence  the 
greater  the  number  of  atoms  in  a  body,  the  greater  in  the  same 
ratio  is  the  attraction  exerted  upon  it.     That  gravity  varies  as  the 
mass  is  also  proved  from  the  observed  fact,  that  in  a  vacuum  it 
gives  the  same  velocity,  in  the  same  time,  to  every  mass,  however 
great  or  small,  and  of  whatever  species  of  matter.     The  greater 
the  mass,  the  greater  the  force  must  be  to  give  to  it  the  same 
velocity  (Art.  12). 

If  a  body  is  not  free  to  move,  its  tendency  towards  the  earth 
causes  pressure;  and  the  measure  of  this  pressure  is  called  the 
weight  of  the  body.  t 

Weight  and  mass  must  not  be  confused.  The  weight  of  a  body 
•depends  upon  its  neighborhood  to  the  earth.  In  free  space  it 
would  be  zero.  The  mass  remains  constant,  wherever  in  the 
universe  it  may  be.  Representing  the  force  exerted  by  gravity 
-u pon  a  body  by  G,  and  its  weight  by  W,  we  have 
G  oc  m  ;  and  w  cc  m. 

Here  G  represents  the  total  force  exerted  by  the  earth  upon  a 
body.  When  gravity  is  acting  upon  1  gram  of  matter,  G  —  981 
dynes.  If  2  grams  were  subjected  to  its  pull,  G  =  2  x  981 
<lynes. 

16.  Relation  of  Gravity  and  Distance.— At  different  dis- 
tances above  the  earth's  surface,  gravity  varies  inversely  as  the  square 
of  the  distance  from- the  centre. 

The  demonstration  of  this  proposition  is  reserved  for  astron- 


10  MECHANICS. 

omy,  where  it  is  shown  by  the  movements  of  the  bodies  in  the 
solar  system  that  this  law  applies  to  them  all. 

The  moon  is  GO  times  as  far  from  the  earth's  centre  as  the  dis- 
tance from  that  centre  to  the  surface  :  therefore  the  attraction  of 
the  earth  upon  the  particles  of  the  moon  is  3600  times  less  than 
upon  particles  at  the  surface  of  the  earth.  At  the  height  of  4000 
miles  above  the  earth,  gravity  is  four  times  less  than  at  the  sur- 
face. But  the  heights  at  which  experiments  are  commonly  made 
upon  the  weights  of  bodies  bear  so  small  a  ratio  to  the  radius  of 
the  earth,  that  this  variation  is  commonly  imperceptible.  At  the 
height  of  half  a  mile,  the  diminution  does  not  amount  to  more 
than  about  TTiV?^h  Par^  °f  the  weight  at  the  surface.  For,  let 
r  =  the  radius  of  the  earth  =  4000  miles,  nearly  ;  and  let  x  be 
the  height  of  the  body,  w  its  weight  at  the  earth's  surface,  and 
10'  its  weight  at  the  height  x.  Then, 

r*  :  :  ra  +  2r.r  +  x*  :  r". 


But  when  x  is  a  small  fraction  of  r,  x*  may  be  neglected,  and 
the  formula  becomes  w  —  w  =  --  ~     .......     (/>'). 


Let  x  be  half  a  mile  ;  then  -        —  -  =  i-gVyth  part  of  the  whole 

weight  ;  or,  a  body  would  weigh  so  much  less  at  the  height  of  half 
a  mile  than  at  the  surface  of  the  earth.  But  if  the  height  were  as 
great  as  100  miles  above  the  earth,  the  loss  should  be  calculated 
by  formula  (A),  since  the  other  would  give  a  result  too  small  by 
one  per  cent,  or  more,  according  to  the  height. 

What  loss  of  weight  would  a  body  sustain  by  being  elevated  500 
miles  above  the  earth  ?  Am.  -J-J,  or  more  than  1  of  its  weight. 

The  relation  of  gravity  to  distance  is  expressed  by  the  formula 

G  oc   —  ;  and  as  G  oc  m  also,  it  varies  as  the  product  of  the  two  ; 

that  is,  G  oc  —  ;  or  gravity  towards  the  earth  varies  as  the  mass  of 
a 

the  body  directly,  and  as  the  square  of  the  distance  from  the  earth'* 
centre  inversely. 

17.  Gravity  within  a  Hollow  Sphere.  —  A  particle  situated 
within  a  spherical  shell  of  uniform  density  is  equally  attracted  in 
all  directions,  and  remains  at  rest.  This  is  true,  because,  in  every 
direction  from  the  particle,  the  mass  varies  at  the  same  rate  as  the 
square  of  the  distance,  so  that  attraction  increases  for  one  reason, 
as  much  as  it  diminishes  for  the  other  ;  which  is  proved  as  follows  t 


GRAVITY    WITHIN    A    SOLID    SPHERE. 


11 


FIG.  1. 


Let  the  particle  P  (Fig.  1)  be  at  any  point  within  the  spherical 
shell  A  B  C  D.  Let  two  opposite  cones  of  revolution,  of  very 
small  angle,  have  their  vertices  at  P,  and  suppose  the  figure  to  be 
a  section  through  the  centre  of  the  sphere  and  the  axis  of  the 
cones.  Then  A  B  and  a  b  will  be  the  major  axes  of  the  small 
ellipses,  which  are  the  bases  of  the  cones,  and 
which  may  be  considered  as  plane  figures.  By 
geometry,  A  P  :  P  B  ::  P  b  :  P  a  ;  and  the 
angles  at  P  being  equal,  the  triangles  are  simi- 
lar ;  hence  the  angles  B  and  a  are  equal. 
Therefore,  the  bases  of  the  cones  are  similar 
ellipses,  being  sections  of  similar  cones,  equally 
inclined  to  the  sides.  By  similar  triangles, 
A  P*  '•  P  b"  '•  '•  A  B*  '•  a  b1.  Let  m  and  m  repre- 
sent the  masses  of  the  thin  laminae  which  form  the  bases ;  then, 
since  similar  ellipses  are  to  each  other  as  the  squares  of  their  major 
axes,  we  have  from  the  above  proportion 

a a  m  m' 

m  :  m'  ::  AP  :Pb,  or  =-,  =  =r 

But and  ^=^  represent  the  attractions  of  the  bases  respec- 
tively on  the  particle  (Art.  16)  ;  and  since  these  are  equal,  the  par- 
ticle is  equally  attracted,  but  in  opposite  directions,  by  the  two 
elliptical  segments  of  the  shell,  and  remains  at  rest.  The  cones  of 
revolution  may  be  multiplied  until  their  bases  cover  the  whole 
sphere.  In  each  case  the  attractions  of  the  opposite  segments  neu- 
tralize each  other  and  the  particle  remains  at  rest. 

18.  Gravity  within  a  Solid  Sphere. — "Within  a  solid  sphere 
of  uniform  density,  weight  varies  directly  as  the  distance  from  the 
centre. 

Let  a  particle  P  (Fig.  2)  be  within  the  solid 
sphere  of  A  D  C  ;  and  call  its  distance  from  the 
centre  d.  Now,  by  the  preceding  article  the 
shell  exterior  to  it,  ADR,  exerts  no  influence 
upon  it,  and  it  is  attracted  only  by  the  sphere 
PUS.  Let  m  represent  the  mass  of  this 

re  ;  then  gravity  varies  as  — rp    But  m  oc  d* ; 


FIG.  2. 


,>*„ 


d.  Hence,  in  the  earth  (if  it  be  supposed  spher- 
ical and  uniformly  dense,  though  it  is  neither  exactly),  a  body  at 
the  depth  of  1000  miles  weighs  three-fourths  as  much  as  at  the  sur- 


12  MECHANICS. 

face,  and  at  2000  miles  it  weighs  half  HIS  much,  while  at  the  centre 
it  weighs  nothing. 

Comparing  this  proposition  with  Art.  16,  we  learn  that  just  at 
the  surface  of  the  earth  a  body  weighs  more  than  at  any  other  place 
without  or  within.  Within,  the  weight  diminishes  nearly  as  the 
distance  from  the  centre  diminishes  ;  without,  it  diminishes  as  the 
square  of  the  distance  from  the  centre  increases. 

At  the  surface  of  spheres  having  the  same  density,  weight  varies 
as  the  radius  of  the  sphere.  Let  r  be  the  radius  of  the  sphere,  and 

in  r* 

m  its  mass  ;  then,  since  G  oc  — ,  in  this  case   it  varies   as  — 7-  oc   r. 
r-  r~ 

Therefore,  if  two  planets  have  equal  densities,  the  weight  of  bodies 
upon  them  is  as  their  radii  or  their  diameters.  If  a  ball  two  feet 
in  diameter  has  the  same  density  as  the  earth,  a  particle  of  dust  at 
its  surface  is  attracted  by  it  nearly  21  millions  of  times  less  than  it 
is  by  the  earth. 

19.  Questions  for  Practice  — 

1.  How   much  weight   would  a  rock   that    weighs    ten    tons 
(22,400  Ibs.)  at  the  level  of  the   sea,  lose  if   elevated  to  the  top  of 
a  mountain  five  miles  high?  Ans.   55.8952  Ibs. 

2.  If  the  earth  were  a  lidllow  sphere,  and  if,  through  a  hole 
bored  through  the  centre,  a   man  were  let  down  by  a  rope,  would 
the  force  required  to  support  him  be  increased  or  diminished  as  he 
descended  through  the  solid  crust,  and  where  would  it  become  equal 
to  nothing  ? 

3.  How  much  would  a  44-pound  shot  weigh  at  the  centre   of 
the  earth  ;  how  much  at  a  poiut  half-way  from  the  centre  to  the 
surface  ;  and  how  much  100  miles  below  the  surface  ? 

4.  If  a  ball  of  the  same  density  with  the  earth,  -j'yth  of  a  mile 
in  diameter,   were  to  fall  through  its  own  diameter  toward  the 
-earth,  what  space  would  the  earth  move  through  to  meet  the  ball, 
the  diameter  of  the  earth  being  taken  at  8000  miles  ? 

Ans.  imFTnri-7nnnnr  inch,  nearly. 

5.  If  a  hole  were  bored   through  the  centre  of  the  earth,  what 
would  be  the  conditions  of  the  motion  of  a  stone  dropped  into  the 
hole? 

In  its  descent  toward  the  centre,  the  force  of  gravity  would 
continually  decrease  till  at  the  centre^  it  became  zero  ;  but  though 
this  force  decreases  in  intensity,  it  will  at  each  instant  increase  the 
previously  existing  velocity,  though  by  decreasing  increments,  so 
that  the  stone  will  have  its  greatest  velocity  at  the  centre  of  the 
«arth :  it  will  then,  in  an  inverse  order,  suffer  continually  increas- 
ing decrements  of  velocity  until  it  finally  comes  to  rest  at  the  other 
surface  of  the  earth,  when  it  will  return  under  similar  conditions. 


CHAPTER     II. 

VARIABLE    MOTION.— WORK.— ENERGY. 

20.  Relation  of  Time  and  Acquired  Velocity. — When  a 
foody  moves  with  uniform  motion,  s  =  t  v  (Art.  6).     When  a  body 
moves  with  uniformly  varied  motion  the  case  is  somewhat  different. 

Let  us  consider  the  case  of  a  body  that  moves  with  uniformly 
increasing  velocity.  Suppose  the  body  to  start  from  rest  and  at 
the  end  of  the  1st  second  to  have  acquired  a  velocity  of  10  cm.  per 
second  ;  that  is  to  say,  a  velocity  which  would  carry  it  over  10  cm. 
per  second  during  the  next  and  each  succeeding  second,  if  the 
force  ceased  to  act  at  the  end  of  the  first  second.  Now,  since  the 
velocity  is  supposed  to  increase  uniformly,  we  shall  have  at  the  en-1 
of  the  2d  second  a  velocity  of  20  cm.,  at  the  end  of  the  3d  a 
velocity  of  30  cm.,  and  so  on. 

Hence  the  first  law  of  motion  under  the  action  of  a  constant 
force :  In  uniformly  accelerated  motion  Uie  acquired  velocities  vary 
as  the  limes. 

21.  Space  Passed  Over. — Since  the  body  started  from  rest 
and  gained  uniformly  in  velocity  till  it  acquired  a  velocity  of  50  cm. 
per  second  at  the  end  of  the  5th  second,  it  is  evident  that  its  aver- 
age velocity  was  25  cm.  per  second  ;  for  at  the  start  its  velocity  was 
0  and  at  the  end  was  50  ;  at  an  interval  of  one  second  after  starting 
it  had  a  velocity  of  10,  and  one  second  before  the  end  of  the  time 
considered  it  had  a  velocity  of  40  ;  two  seconds  after  starling  the 
velocity  was  20,  and  two  seconds  before  the  end  (if  the  tune  the 
velocity  was  30.     Thus  the  less  velocity  at  any  given  interval  is  bal- 
anced by  the  greater  velocity  during  the  corresponding  interval  of 
the  pair,  and  we  are  thus  enabled  to  find  the  distance  passed  over 
by  multiplying  the  average  velocity,  of  25  cm.  per  second,  by  the 
time,  5  seconds,  giving  the  space  125  cm. 

22.  Space  Described  during  ist  Second.— We  have  con- 
sidered the  velocities  at  intervals  of  one  second,  but  we  could  have 
chosen  smaller  intervals  as  well,  and  no  matter  how  small  we  malce 
our  unit  of  time,  the  law  holds  good.     Now,  during  the  first  sec- 
ond the  body  acquired  a  velocity  of  10  cm.,  and  if  we  suppose  the 
first  second  to  be  divided  into  10  equal  intervals,  we  may  apply  the 


MECHANICS. 


same  analysis  to  these  as  to  the  five  full  seconds  already  considered  ; 


and  we  find  the  average  velocity  to  be 


10  +  0, 


or  5  cm.  :   hence, 


since  the  body  moved  for  one  second  with  a  velocity  which  would 
average  5  cm.  per  second,  it  must  have  moved  over  5  cm.  :  hence 
a  body  starting  from  rest  will,  under  the  action  of  a  constant  force, 
move  during  the  first  second  over  a  space  equal  to  one-half  the  velocity 
acquired  at  t/ie  end  of  that  second. 

23.  Space    Described   during   any  Second. — The    space 
described  during  any  second  is  one-half  the  velocity  impressed  upon 
the  body  by  the  constant  force  during  that  second,  plus  the  space 
described  by  reason  of  velocity  already  impressed  upon  the  body- 
by  previous  action  of  the  force. 

24.  Relations  of  Time,  Space,  and  Acquired  Velocity. 
— It  is  necessary  to  know  all  the  possible  relations  between  the 
space,  time,  and  acquired  velocities.     Let  us  now  examine  the  re- 
lations between  time  and  space.     During  the  first  second  the  body, 
in  the  case  already  given,  moves  over  5  cm.  and  acquires  a  velocity 
of  10  cm.  ;  during  the  2d  second  it  will  move  over  10  cm.  in  con- 
sequence of  the  velocity  already  impressed,  and  over  5  cm.  addi- 
tional because  of  the  continued  action  of  the  force,  making  a  total 
of  15  cm.    At  the  beginning  of  the  3d  second  the  velocity  is  20  cm., 
and  the  body  will  move  over  20  cm.  in  consequence  of  this,  together 
with  5  cm.  more  on  account  of  the  c6ntinued  action  of  the  force  ; 
and  so  on  to  the  end  of  the  time. 

Hence,  we  have — 


Times. 

Ac.  vel.  at  begin- 
ning of  interval. 

Spaces  described 
during  interval. 

Total  spaces. 

1st  sec. 

0 

5 

For  1  sec. 

5 

2d     " 

10 

15 

«     2     " 

20 

3d     " 

20 

25 

"     3     " 

45 

4th    " 

30 

35 

«    4    '" 

80 

6th    " 

40 

45 

"     5     " 

125 

Examining  the  above  results,  and  calling  the  space  described 
during  the  1st  second  S,  we  have  the  space  during 

2  sec.  =  S  x     4  =  8   x   22 

3  "     =  S  x     9  =  £  x   3a 

4  "     =  S  x   16  =  S  x   4' 

5  "     =  S   *   25  =  8  x   5a 

That  is  to  say,  The  spaces  described  under  the  action  of  a  constant 


UNIFORMLY    ACCELERATED    MOTION.  15 

force  are  proportional  to  the  squares  of  the  times  during  which  the 
force  acts. 

Acquired  velocities  are  proportional  to  the  times,  and  there- 
fore the  spaces  must  be  also  proportional  to  the  squares  of  the 
acquired  velocities. 

25.  Laws  of  Uniformly  Accelerated  Motion. — To  re- 
capitulate ;  when  bodies  move  under  the  action  of  a  constant  force, 
the  following  relations  exist  between  space,  time,  and  velocity  : 

1.  The  acquired  velocities  vary  as  the  times. 

2.  The  spaces  vary  as  the  squares  of  the  times. 

3.  TJie  spaces  vary  as  the  squares  of  the  acquired  velocities. 

As  an  aid  to  the  memory,  the  following  analogy  may  be  em- 
ployed. Let  s  be  the  space  described,  v  the  velocity  acquired  by 
a  body  moving  from  rest  for  the  time  t,  s'  the  space  described,  v' 
the  velocity  acquired  at  any  other  period  t'  ;  then,  from  what  has 
already  been  demonstrated,  if  t  and  t'  be  represented  by  the  lines 
A  B  and  A  D  (Fig.  3),  and  v  and  v'  by  the  lines  B  G  plo_  3. 

and  D  E,  drawn  at  right  angles  to  them,  s  and  A. 
3'  will  be  represented  by  the  triangles  ABC  and 
A  DE.  ForAEC:ADE::All*:AD3;  or 
as  B  C2  :  D  E3  ;  or  s  :  s'  ::  f  :  t'"  •  or  as  u*  :  v"J. 
The  velocity  acquired  varies  as  the  time  ;  from 
the  similar  triangles  A  B  C,  A  D  E,  we  have  B  C 
I  D  E  : :  A  B  :  A  D,  or,  -o  :  v'  : :  t  :  t'. 

•  26.  Formulae. — Let  us  represent  by/  the  acceleration  due  to  a 
force,  that  is  to  say,  the  increase  of  velocity  per  second  due  to  the 
action  of  the  force  ;  then  the  space  passed  over  during  the  1st 
second,  if  starting  from  rest,  would  be  £  /,  as  deduced  in  Art.  22. 
Calling  the  total  space  s,  time  in  seconds  t,  and  acquired  velocity  v, 
we  have,  from  the  above  laws, 

«   =     fl, 
s    =  lft\ 
and      v*  —  2/s. 

27.  Applications   of  the  Formulae. — 

1.  Find  expressions  for  /  in  terms  of  v  and  t,  s  and  t,  v  and  s ; 
for  t  in  terms  of  f  and  s  ;  for  v  in  terms  of  s  and  t. 

[The  acceleration  produced  by  gravity,  acting  for  one  second 
upon  any  freely  falling  body,  no  matter  what  its  mass,  is  981  cm., 
or  32.2  ft.  This  is  generally  represented  by  g  and,  when  solving 
problems  concerning  falling  bodies,  g  may  be  substituted  for  f  in 
the  formulas  of  Art.  26.] 


16  MECHANICS. 

2.  A  body  falls  10  seconds  :  Requited  (a)  the  velocity  acquired  ; 
(b)  whole  distance  fallen  through. 

.  .  f  98.1  m.  per  second.  (  490.5  in. 

An*,  (a)  I  322  ft.     «        «          (^  }  1610  ft. 

3.  A  body  has  fallen  through  90  meters  :  Required  (a)  the  timo 
of  falling  ;  (b)  the  final  velocity. 

4.  A  body  falls  4  seconds  and  acquires  a  velocity  of  300  feet. 
What  was  the  acceleration,  and  what  space  was  passed  over  ? 

Ans.f=  75  ft;  s  =  COO  ft. 

5.  A  body  falls  through  402.5  ft.:  Required  (a)  time  of  falling  ; 
(b)  acquired  velocity  ?  -4ns.  (a)  5  sec. ;  (b)  161  ft. 

28.  Uniform  and  Uniformly  Varied  Motion  Combined. 
— Thus  far  we  have  assumed  the  body  to  start  from  rest,     If  the 
condition    be  changed  and  the   body  be  considered  as  having   a 
uniform  motion  at  the  time  the  action  of  tile  constant  force  begins, 
we  have  merely  to  combine  the  formula  for  that  motion  with  that 
of  uniformly  accelerated  motion  already  used.     Thus,  if  a  body  is 
thrown  downward  with  a  force  which  gives  it  a  velocity  of   10  m. 
per  second,  how  far  will  it  fall  in  4  seconds,  and  what  velocity  will 
it  have  at  the  end  of  that  time  ?     Under  the  action  of  the  down- 
ward impulse  alone,  it  would  move  over  4  x  10  m.  =  40  iru     Un- 
der the  action  of  gravity  it  would   move  over  4.9x4'  =  78.4m.; 
combining  these  two  effects,  we  have  118.4  m.  as  the  total  distance 
passed  over  in  the  given  time.  Designating  the  velocity  due  to  the  im^ 
pulse,  usually  called  the  "  initial  velocity,"  by  v,  we  have  total  space 
S  =  v  t  -t-  £  g  f  ;  and  also  final  velocity  v'  =  v  +  gt  =  lQ  +  39.2  ^ 
49.2  m.  per  second.  • 

29.  Uniformly  Retarded  Motion. — In  like  manner  we  can 
determine  the  results  when  a  constant  force  acts  to  retard  velocity 
already  imparted,  by  merely  taking  the  difference  of  the  two  effects. 

A  body  receives  an  impulse  of  1 00  m.  per  second,  and  is  retarded 
by  a  constant  force  whose  acceleration  is  10  m.  per  second  ;  how 
far  will  the  body  move  in  5  seconds  ?     We  now  have 
8=vt—  \fil  ; 

=  100  x  5  —  5  x  25  =  375. 

30.  Space  in  any  Given  Second  or  Seconds  of  Fall. — 
If  it  be  required  to  find  how  far  a   body  will  move  during   any 
specified  unit  or  units  of  time,  proceed  thus  :  suppose  it  to  be  re- 
quired to  determine  how  far  the  body  will  move  during  the  7th 
second  ;  for  the  whole  7  seconds, 

«  =  $/f  =  i/x  T; 
for  six  seconds,  «'  —  \  ft'*  —  \f  x  6J  ; 

>  -  6')  -  *£ 


QUESTIONS    ON    FALLING    BODIES.  \f 

Suppose  we  are  required  to  determine  the  space  described  dur- 
ing the  last  three  seconds  : 

s  =  i//'   =£/  x  T; 


1.  How  far  does  a  body  move  in  the  14th  second  of  its  fall  ? 

Ans.  434.7  ft. 

2.  A  body  had  been  falling  2  minutes  ;  how  far  did  it  move  in 
the  last  second  ?  Ans.  3847.9ft. 

3.  What  space  was  described  in  the  last  two  seconds  by  a  body 
which  had  fallen  300  feet  ?  Ans.  214.1  ft. 

4.  A  body  had  been  falling  8£  seconds  ;  how  far  did  it  descend 
in  the  next  second  ?  Ans.  289.8ft. 

3L  Questions  on  Falling  Bodies. — 

[</  —  9.8  m.] 

1.  A  stone   is  thrown  vertically  upward  with  a  velocity  of  100 
meters.     When  would  it  return  to  its  original  position  ? 

Ans.  20.4  sec. 

[A  little  consideration  will  show  that  a  body  projected  vertically 
upwards  will  occupy  the  same  time  in  ascending  as  in  descending. 
The  acquired  velocity  upon  its  return  is  equal  to  the  initial  veloc- 
ity.] 

2.  A  stone  is  thrown   into  a  pit  150  m.  deep  and  reaches  the 
bottom  in  4  seconds  :  Required   (a)  initial  velocity  ;  (b)  acquired 
velocity.  Ans.   (a)  17.9  ;  (b)  57.1  m.  per  sec. 

3.  A  cannon-ball  has  been  shot  vertically  upward  with  a  veloc- 
ity of  250  meters  in  a  second.     After  what  interval  of  time  would 
its  velocity  have  been  reduced  to  54  meters  under  the  retarding 
influence  of  gravity,  and  what  space  would  have  been  traversed  by 
the  ball  at  the  end  of  this  time?  Ans.  20  sec.  ;  3040  m. 

4.  A  stone  is  thrown  from  a  balloon  with  a  velocity  of  50  me- 
ters in  a  second.     How  soon  will  the  velocity  amount  to  99  me- 
ters in  a  second,  and  through  what  distance  will  the  stone  have 
fallen  ?  Ans.  5  sec.  ;  372.5  m. 

5.  How  far  would  a  body  go  in  the  10th  second  of  its  fall? 

6.  A  body  has  acquired  in  falling  a  velocity  of  73.5  meters  per 
second  :  Required  (a)  the  time  of  falling  ;  (b)  the  distance  fallen 
through.  Ans.  (a)  1%  sec.  ;  (b)  275.6  m. 

7.  A  body  in  falling  passed  over  44.1  meters  in  the  last  second  : 
Required  (a)  the  time  of  falling  ;  (b)  the  distance  fallen. 

[g  =  32.2ft] 
2 


IS  MECHANICS. 

8.  An  archer  •wishing  to  know  the  height  of  a  tower/ found  that 
an  arrow  sent  to  the  top  of  it  occupied  8  seconds  in  going  and  re- 
turning ;  what  was  the  height  of  the  tower?  Aiis.  257.6  ft. 

9.  In  what  time  would  a  man  fall  from  a  balloon  three  miles 
high,  and  what  velocity  would  he  acquire  ? 

Ans.  t  =  31.4  sec.  ;  v  =  1011.1  ft. 

10.  A  body  having  fallen  for  3£    seconds,  was  afterward    ob- 
served to  move  with  the  velocity    which  it  had  acquired  for   2^ 
seconds  more  ;  what  was  the  whole  space  described  by  the  body  ? 

Ans.  478.9  ft.,  very  nearly. 

11.  Through  what  space  would   the  aeronaut  (in  Question  9) 
fall  during  the  last  second  ?  Ans.  995  ft. 

12.  A  body  has  fallen  from  the  top  of  a  tower  340  feet  high  ; 
what  was  the  space  described  by  it  in  the  last  three  seconds  ? 

.4ns.  299.5  ft. 

13.  Suppose  a  body  be  projected  downward  with  a  velocity  of 
18  feet  in  a  second  ;  how  far  will  it  descend  in  15  seconds  ? 

Ans.  3892.5  ft. 

14.  A  body  is  projected  upward  with  a  velocity  of  65  feet  in  a 
second  ;  how  far  will  it  rise  in  two  seconds?  Ans.  65.6  ft. 

15.  With  what  velocity  must  a  stone  be  projected  into  a  well 
450  feet  deep,  that  it  may  arrive  at  the  bottom  in  four  seconds  ? 

Ans.  48.1  ft.  in  a  second. 

16.  The  space  described  in  the  fourth  second  of  a  fall  was  to  the 
space  described  in  the  last  second  except  four,  as  1  :  3  ;  what  was 
the  whole  space  described  by  the  body  ?  Ans.  3622.5  ft 

17.  A  staging  is  at  the  height  of  84  ft.  above  the  earth.     A  ball 
thrown  upward  from  the  earth,  after  an  absence  of  7  seconds,  fell 
on  the  staging  ;  what  was  the  velocity  of  projection  ? 

Ans.  124.7  ft.  per  second. 

18.  A  body  is  projected  upward  with  a  velocity  of  483  feet  in 
#  second  ;  in  what  time  will  it  rise  to  a  height  of  1610  feet  ? 

Ans.  t  =  3.8  sec.,  or  26.2  sec. 

19.  From  a  point  386.4  feet  above  the  earth  a  body  is  projected 
upward  with  a  velocity  of  161  feet  in  a  second  ;  in  what  time  will 
it  reach  the  surface  of  the  earth,  and  with  what  velocity  will  it 
strike?  Ans.  t  =  12  sec.,  v  =  225.4  ft. 

20.  A  body  is  projected  upward  with  a  velocity  of  64.4  feet  in  a 
second  ;  how  far  above  the  point  of  projection  will  it  be  at  the  end 
of  4  seconds?  Ans.  0  ft 

21.  A  body  is  projected  upward  with  a  velocity  of  128.8  feet  in 
&  second  ;  where  will  it  be  at  the  end  of  10  seconds? 

Ans.  322  ft.  below  the  point  of  projection. 


ATWOOD'S    MACHINE. 


19 


32.  Atwood's  Machine. — We  have  seen  that  gravity,  acting 
for  a  given  time,  gives  the  same  velocity  to  all  bodies,  heavy  or 
light.  In  acting  upon  the  heavy  one,  however,  it  employs  more 
force,  and  if  this  same  force  were  em-  FIG.  4. 

ployed  in  moving  a  still  heavier  body, 
it  would  give  it  a  smaller  velocity. 

Now  the  velocities  of  freely  falling 
bodies  are  so  great  that  we  cannot  ex- 
periment upon  them  directly  in  ascer- 
taining their  laws.  We  can,  however, 
make  the  force,  exerted  by  gravity,  on 
a  given  mass  (say  1  gram),  cause  the 
motion  of  a  heavier  known  mass  (100 
grams),  and  thus  reduce  the  velocities 
produced,  in  a  given  ratio  (y-J-o).  The 
smaller  velocities  can  then  be  'readily 
observed  and  measured. 

Employing  this  principle,  Atwood 
has  constructed  a  machine  by  which  all 
the  facts  of  uniformly  accelerated  or 
retarded  motion  can  be  illustrated  with 
sufficient  accuracy. 

This  machine  is  represented  in  Fig. 
4.  From  the  base  of  the  instrument, 
which  is  furnished  with  leveling  screws, 
rises  a  substantial  pillar,  about  seven 
feet  high,  supporting  a  small  table  upon 
the  top. 

Above  the  table  is  a  grooved  wheel, 
delicately  suspended  on  friction-wheels, 
and  protected  from  dust  by  a  glass  case. 
Two  equal  poises,  M  and  M',  arS  at- 
tached to  the  ends  of  a  fine  cord,  which* 
passes  over  the  groove  of  the  wheel.  As 
gravity  exerts  equal  forces  on  M  and 
M',  they  are  in  equilibrium.  To  set 
them  in  motion,  a  small  bar,  ra,  is  placed 
on  M,  which  will  immediately  begin  to 
descend,  and  M'  to  rise.  But  this  mo- 
tion will  be  slower  than  in  falling  freely, 
because  the  force  which  gravity  exerts 

on  the  bar  must  be  communicated  to  the  poises,  and  also  to  the 
revolving  wheel  over  which  the  cord  passes.  By  increasing  the 
poises  M,  M,  and  diminishing  the  bar  m,  the  motion  may  be 


20  MECHANICS. 

made  as  slow  as  we  please.  0  is  a  simple  clock  attached  to  the- 
pillar  for  measuring  seconds,  and  for  dropping  the  poise  31  at  the 
beginning  of  a  vibration  of  the  pendulum.  Q  is  a  scale  of  cen- 
timeters or  inches  extending  from  the  base  to  the  table.  The 
stage  A  may  be  clamped  to  any  part  of  the  scale,  in  order  to  stop 
the  poise  Mm  its  descent,  as  represented  at  C.  The  rings-S,  which 
is  large  enough  to  allow  the  poise,  but  not  the  bar,  to  pass  through 
it,  is  also  clamped  to  the  scale  wherever  the  acceleration  is  to  cease. 

Let  M  be  raised  to  the  top,  and  held  in  place  by  a  support,  and 
then  let  the  pendulum  be  set  vibrating.  When  the  index  passes 
the  zero  point  the  clock  causes  the  support  to  drop  away,  and  the 
poise  descends.  The  pendulum  shows  how  many  seconds  elapse 
before  the  bar  is  arrested  by  the  ring,  and  how  many  more  before 
the  poise  strikes  the  stage.  From  the  top  to  the  ring  the  motion 
is  accelerated  by  the  constant  fraction  of  gravity  acting  on  it ;  from 
the  ring  to  the  stage  the  poise  moves  uniformly  with  the  acquired 
velocity.  Moreover,  the  resistance  of  the  air  is  so  much  diminished 
when  the  motion  is  slow,  that  a  good  degree  of  correspondence  is 
found  to  exist  between  the  experiments  and  the  results  of  calcula- 
tion. 

If  we  disregard  the  mass  of  the  wheel  as  not  sensibly  affecting 
the  results,  which  we  may  do  in  practice  if  the  weights  are  heavy 
as  compared  with  it,  we  may  illustrate  the  action  of  the  machine  by 
the  following  case  :  Suppose  M  and  M'  to  weigh  together  99  grams 
and  ??i  to  weigh  1  gram.  The  force  exerted  on  1  gram  must  move 
100  grams,  and  hence  the  velocity  will  be  but  y^  as  great  as  by 
free  falling.  The  acquired  velocity  would  then  be  but  9.8  cm.  at 
the  end  of  the  first  second,  and  would  pass  over  but  4.9  cm.  during 
the  first  second.  Such  a  small. velocity  can  be  readily  observed. 

Ex.  1.  If  M  =  M'  =  24.5  grams,  and  m  =  1  gram  what  is  the 
acceleration  ? 

33.  ^Af  ork. —  Work  is  the  production  of  motion  against  resistance* 

Whenever  a  body  is  set"  in  motion  against  the  restraining  influ- 
ence of  any  force,  work  is  said  to  be  performed.  If  no  motion  be 
produced,  then  no  work  is  performed.  If  there  be  no  opposing 
force,  no  work  is  performed. 

Whenever  a  weight  is  lifted  from  the  ground,  work  is  done 
against  gravity  ;  the  same  is  true  when  a  hill  or  pair  of  stairs  are 
ascended.  Winding  up  a  watch  requires  work  against  the  force  of 
elasticity.  Turning  a  grindstone  demands  work  against  friction. 

Let  A  represent  the  work  done,  F  the  force  performing  the- 
work,  and  s  the  distance  moved  in  the  direction  of  this  force,  them 
work  is  calculated  by  the  formula, 

A  =  Fs. 


ENERGY.  21 

In  order  to  get  a  unit  for  work  we  must  make  F  and  s  each 
equal  to  unity.  This  unit  of  work,  called  The  Erg,  is  the  work  per- 
formed by  the  force  1  dyne  in  moving  a  body  1  centimeter  in  the 
direction  it  is  acting. 

This  unit  is  very  small  (the  force  exerted  by  gravity  on  one 
gram  =  981  dynes),  and  hence  the  Megalerg  (=  1,000,000  ergs)  is 
used.  The  practical  unit,  which  is  still  much  used,  the  Foot-pound, 
takes  the  force  exerted  by  gravity  on  a  pound  of  matter  for  F,  and 
the  foot  for  unit  s.  It  is  often  convenient,  in  measuring  work,  to 
consider  the  work  done  as  equal  to  the  product  of  the  force  which 
opposes  motion,  and  the  motion  produced  resolved  in  the  direc- 
tion of  this  opposing  force.  This  is  especially  the  case  when 
gravity  is  the  opposing  force.  Thus  the  force  used  in  raising  a 
weight  is  in  a  vertically  upward  direction — the  resisting  force, 
gravity,  working  vertically  downward.  The  work  performed  equals 
the  pull  of  gravity  times  the  vertical  distance  moved  through.  No 
more  work  would  be  performed  in  climbing  a  ladder  to  the  top  of 
a  precipice  than  by  climbing  the  hill  to  the  rear  of  it.  Thus  if 
gravity  be  supposed  to  work  in  the 

direction     A   B     (Fig.    5),     the    same  A 

amount  of  work  will  be  performed  in 
rising  to  A,  by  each  of  the  paths  B  A, 
C  A,  D  A,  and  E  A. 

In  practical  life,   though    the   hod- 
carrier    and    telegraph  -  pole     climber 
would  appear  to  have  much  work  to  per-   E          # <; 
form,  the  ditch  digger  is  said  to  do  the  most  work  in  a  day. 

Ex.  1.  How  much  work  is  performed  in  raising,  a  liter  of  water 
1m?  Ans.  98.1  megalergs. 

Ex.  2.  A  plank  5  feet  long  reaches  from  the  threshold  of  a  ware- 
house door  to  a  plate  on  the  ground  4  feet  from  the  building. 
What  work  is  performed  in  rolling  a  333^  pounds  cask  from  the 
ground  into  the  warehouse  ?  Ans.  1000  foot-pounds. 

34.  Energy. — Energy  is  the,  capacity  to  do  work.  The  energy 
possessed  by  bodies  is  divided  into  two  classes. 

1.  Potential  energy  is  the  energy  which  a  body  has  in  virtue  of  its 
position.     The  weights  of  a  clock  can  do  work   in  virtue  of  their 
position   relative  to  the  earth.     Water  at  a  high  level  also  has  po- 
tential energy.     A  wound-up  spring  has  energy  from  the  strained 
position  of  its  molecules. 

2.  Kinetic  energy  is  the  energy  which  a  body  has  in'  virtue  of  its 
motion.     A   rifle-bullet  because  of  its  velocity  has  energy,  and  by 
proper  mechanism  could  be  made  to  raise  a  weight.     In  fact,  if 


22  MECHANICS. 

fired  vertically  upward,  it  not  only  does  work  in  lifting  itself,  but, 
if  the  distance  it  rises  and  its  weight  are  known,  we  can  measure 
the  work  it  does. 

35.  Transmutation  of  Energy. — The  rifle-bullet,  in  its  as- 
cent, gradually  loses  its  kinetic  energy  until,  at  the  highest  point, 
it  possesses  none.  What  has  become  of  it?  Having  lost  its 
motion  it  now  possesses  potential  energy  because  of  its  position 
relative  to  the  earth.  Falling  backward  it  transmutes  its  potential 
into  kinetic  energy  until,  arriving  at  the  starting-point,  it  has  no 
potential  but  all  kinetic  energy.  During  all  the  time  of  its  flight 
its  total  energy,  kinetic  plus  potential,  was  constant,  and  it  returns 
with  the  same  velocity  that  it  started  with.  A  FIG.  6. 

swinging  pendulum  also  illustrates  the  trans- 
mutation of  energy.  At  either  extremity  of  its 
swing  (Fig.  6),  its  energy  is  all  potential.  At  the 
lowest  point  of  the  arc  it  is  all  kinetic.  At  in- 
termediate points  it  possesses  both  kinds. 


36.  Calculation  of  Energy.  —  Potential 
energy  is  possessed  by  a  body  because  work  has  been  performed 
upon  it.  It  is  capable  of  performing  the  same  amount  of  work  by 
returning  to  its  former  position.  To  calculate  its  potential  energy 
we  have  only  to  calculate  the  work  necessary  to  bring  it  to  its 
position.  A  mass  m,  raised  vertically  through  s  cm.,  requires  a 
force  of  m  g  dynes,  and  the  work  performed,  that  is,  its 
Potential  energy  =  m  g  s,  ergs. 

Thus  2  grams  raised  1  m.  have  a  potential  energy  =  2  x  981  x 
100  =  186,200. 

In  the  case  of  the  kinetic  energy  of  a  mass  m,  moving  with  a 
velocity  v,  we  must  know  how  much  work  it  can  do  on  itself,  or 
how  high  it  can  raise  itself  vertically.  From  Art.  26  we  know 
that  this  distance, 


Multiplying  both  sides  by  the  acting  force  m  g,  we  have 

.m  g  s  =  £  m  v*  =  kinetic  energy. 

All  the  kinetic  energy  has  been  transmuted  into  potential,  which 
is  measured  by  m  g  s. 

It  is   often  required  to   calculate  the   kinetic  energy  in  foot- 
pounds.    To  do  this  we  have  to  remember  that  weight, 

W=  mg. 
Substituting  this  value  for  m  g  in  the  formula  above  we  have 


a 


DISSIPATION    OF    ENERGY.  23 

If  Wis  measured  in- pounds,  and  v  and  g  in  feet,  the  value  obtained 
is  in  foot-pounds. 

37.  Conservation  and  Dissipation  of  Energy. — One  ;of 

the  most  fundamental  and  important  principles  in  the  whole  sub- 
ject of  physics  is  the  law  of  the  conservation  of  energy.  It  may 
be  stated  as 

The  total  amount  of  energy  in  the  universe  is  constant :  No  energy 
is  ever  lost. 

Whenever  a  certain  amount  of  energy  is  communicated  to 
machinery  by  an  engine,  the  full  amount  is  not  recovered  in  what 
the  machinery  does.  The  rest  is  said  to  be  lost.  Tracing  this  loss 
we  find  a  great  deal  owing  to  friction.  The  overcoming  of  friction, 
however,  produces  heat,  another  form  of  energy.  This  heat  event- 
ually escapes  to  the  earth,  and  there  may  be  of  use  in  furthering 
vegetable  life.  Vegetables  are  possessed  of  potential  energy,  and 
by  their  assimilation  in  the  human  system,  and  combination  with 
the  oxygen  of  the  air,  reappear  as  animal  heat  or  muscular  energy. 
If  we  trace  out  the  course  of  a  certain  amount  of  energy,  we  will 
find  that  it  is  continually  changing  itself  from  one  form  into  another, 
and  eventually  turns  into  heat.  Tracing  its  past  history,  we  find 
that  by  far  the  larger  proportion  came  ultimately  from  the  sun. 
Thus  water  in  a  mill-pond  owes  its  potential  energy  to  the  evapor- 
ating influence  of  the  sun.  The  winds  are  caused  by  the  sun's 
heat,  but  the  tides  largely  by  the  moon's  attraction. 

„  When  a  lead  bullet  is  fired  against  a  stone  wall  its  energy  is 
transformed  into  heat.  This  may  melt  the  lead,  but  finally  the 
heat  goes  into  the  surrounding  objects  and  is  at  a  low  temperature. 
Heat  at  low  temperature  is  of  no  use  to  man  in  performing  woi'k, 
and  hence  the  energy  which  it  represents  is  said  to  be  dissipated, 
Now,  though  the  total  energy  of  the  universe  remains  constant,  the 
amount  which  is  available  to  man  is  decreasing.  Clausius,  con- 
sidering the  ultimate  form  of  energy  to  be  heat,  called  the  expres- 
sion, 

total  quantity  of  heat 

=  entropv, 

average  temperature 

and  expressed  the  above  fact  by  saying,  "  The  entropy  of  the  uni- 
verse approaches  a  maximum." 

38.  Power. — Power  is  the  rate  of  performance  of  work.     If  t 
represents  the  time  required  to  perform  an  amount  of  work  A, 
then  the  power 


24:  MECHANICS. 

There  are  two  practical  units  of  power,  represented  by  the  follow- 
ing equations  : 

w*«..-H2SB_. 

per  second 

33000  ft.-lbs. 

Horse-power  =  —    — : — - — 
per  minute 

One  horse-power  is  equivalent  to  746  Watts. 

39.  Problems  on  Energy. — 

1.  What  is  the  potential  energy  of  25  kilos,  raised  to  a  height 
of  40  meters  ?  Am.  981  x  10"  ergs. 

2.  A  stone,  weighing  6  kilos.,  falls  from  rest,  at  a  place  where 
g  =  980 ;  what  will  be  its  kinetic  energy  at  the  end  of  5  seconds  ? 

Am.  7.203  x  10'"  ergs. 

3.  A  stone  weighing  10  grams,  fired  vertically  upward,  returns 
in  10   seconds ;  when  was  its  kinetic   energy  greatest,    and   how 
much  ?    When  did  it  have  maximum  potential  energy,  and  how 
much  ? 

4.  Find  the  horse-power  of  an  engine  that  should  be  employed 
for  raising  coal  from  a  mine  200  feet  deep,  the  average    daily  (24 
hours)  yield  being  1782  tons.  Ans.  16.8  horse-power. 

5.  A  man  weighing  75  kilos,  can  climb  by  stairs  to  a  floor  20 
meters  above  in  1  minute.     What  is  his  power  ? 


CHAPTER   III. 

COMPOSITION    AND    RESOLUTION    OF    MOTION. 

40.  Motion  by  Two  or  More  Forces. — Motion  produced 
Tby  a  single  force,  either  impulsive  or  continued,  has  been  already 
considered.     But  motion  is  more  generally  caused  by  several  forces 
acting  in  different  directions. 

When  two  or  more  forces  act  at  once  on  a  body,  each  force  is 
called  a  component,  and  the  joint  effect  is  called  the  resultant. 
Forces  may  be  represented  by  the  straight  lines  along  which  they 
would  move  a  body  in  a  given  time  ;  the  lines  represent  the  forces 
in  two  particulars,  the  directions  in  which  they  act  and  their  rela- 
tive magnitudes.  Whenever  an  arrow-head  is  placed  on  a  line,  it 
shows  ia  which  of  the  two  directions  along  that  line  the  force  acts. 

41.  The   Parallelogram  of  Forces. — This  is  the  name 
.given  to  the  relation  which  exists  between  any  two  components 
.and  their  resultant,  and  is  stated  as  follows  : 

If  two  forces  acting  at  once  on  a  body  are  represented  by  the 
•adjacent  sides  of  a  parallelogram,  their  resultant  is  expressed  by 
the  diagonal  which  passes  through  the  intersection  of  those  sides. 

Suppose  that  a  body  situated  at  A  (Fig.  7)  receives  an  impulse 
which,  acting  alone,  would  carry  it 
over  A  B  in  a  given  time,  and  an- 
other winch  would  carry  it  over  A  G 
in  the  same  length  of  time.  If  both 
impulses  are  given  at  the  same  in- 
stant, the  body  describes  A  D  in  the 
same  time  as  A  B  by  the  first  force, 
or  A  C  by  the  second,  and  the  motion  in  A  D  is  uniform. 

This  is  an  instance  of  the  coexistence  of  motions,  stated  in  the 
second  law  of  motion  (Art.  13).  For  the  body,  in  passing  directly 
from  A  to  D,  is  making  progress  in  the  direction  A  C  as  rapidly 
as  though  the  force  A  B  did  not  exist ;  and  at  the  same  time  it 
advances  in  the  direction  A  B  as  fast  as  though  that  were  the 


26  MECHANICS. 

only  force.  When  the  body  reaches  D,  it  is  as  far  from  the  line 
A  B  as  if  it  had  passed  over  AC;  it  is  also  as  far  from  the  line 
A  C  as  if  it  had  gone  over  A  B.  Thus  it  appears  that  both  mo- 
tions A  B  and  A  C  fully  coexist  in  the  progress  of  the  body  along 
the  diagonal  A  D.  That  the  motion  is  uniform  in  the  diagonal 
is  evident  from  the  law  of  inertia ;  for  the  body  is  not  acted  on. 
after  it  leaves  A. 

It  is  evident  that  a  single  force  might  produce  the  same  effect ; 
that  force  would  be  represented,  both  in  direction  and  magnitude, 
by  the  line  A  D.  The  force  A  D  is  said  to  be  equivalent  to  the 
two  forces  A  B  and  A  C. 

42.  Velocities  Represented. — The  lines  A  B  and  A  C  are 
described  by  the  components  separately,  and  the  line  A  D  by  their 
joint  action,  in  the  same  lengtti  of  time.     Hence  the  velocities  in 
those  lines  are  as  the  lines  themselves.     In  the  parallelogram  of 
forces,  therefore,  two  adjacent  sides  and  the  diagonal  between 
them  represent — 

1st.  The  relative  directions  of  the  components  and  resultant  ; 

2d.  Their  relative  magnitudes  ;  and 

3d.  The  relative  velocities  with  which  the  lines  are  described. 

43.  The  Triangle  of  Forces. — For  purposes  of  calculation,. 
it  is  more  convenient  to  represent  two  components  and  their  re- 
sultant by  the  sides  of  a  triangle,  than  by  the  sides  and  diagonal 
of  a  parallelogram.     In  Fig.  7,  C  D,  which  is  equal  and  parallel  to 
A  B,  may  represent  in  direction  and  magnitude  the  same  force 
which  A  B  represents.     Therefore,  the  components  are  A  C  and 
C  D,  while  their  resultant  is  A  D ;  and  the  angle  C  in  the  triangle- 
is  the  supplement  of  CAB,  the  angle  between  the  components- 
Care  should  be  taken  to  construct  the  triangle  so  that  the  sides 
representing  the  components  may  be  taken  in  succession  in  the 
directions  of  the  forces,  as  A  C,  CD;  then  A  D  correctly  repre- 
sents their  resultant.    But,  although  A  C  and  A  B  represent  the 
components,  the  third  side,  C  B,  of  the  triangle  A  C  B,  does  not 
represent  their  resultant,  since  A  C  and  A  B  cannot  be  taken  suc- 
cessively in  the  direction  of  the  forces.    It  is  necessary  to  go  back 
to  A  in  order  to  trace  the  line  A  B.     It  should  be  observed,  that 
though  C  D  represents  the  magnitude  and  direction  of  the  compo- 
nent, it  is  not  In  the  line  of  its  action,  because  both  forces  act 
through  the  same  point  A. 

Three  forces  produce  equilibrium  token  they  may  be  represented 
in  direction  and  magnitude  by  the  three  sides  of  a  triangle  taken 
in  order. 


TRIANGLE  OF    FORCES.  27 

For,  when  three  forces  are  in  equilibrium,  one  of  them  must 
be  equal  to,  and  opposite  to,  the  re- 
sultant of  both  the  others.      But  FlG-  8- 
the  forces  A  C  and  A  B  (Fig.  8) 
produce  the  resultant  A  D  ;  there- 
fore the  equal  and  opposite  force 
D  A,  since  it  is  in  equilibrium  with 
A  D)  is  also  in  equilibrium  with  A  0 
and  A  B,  or  A  C  and  C  D.    Hence 
the  three  forces  A  C,  C  D,  and  D  A,  taken  in  order  around  the 
figure,  produce  equilibrium. 

44.  The  Forces  Represented  Trigonometrically. — Since 
the  sides  of  a  triangle  are  proportional  to  the  sines  of  the  opposite 
angles,  these  sines  may  also  represent  two  components  and  their 
resultant.    Thus,  the  sine  ofCAD  corresponds  to  the  component 
A  B  (=  C  D) ;  the  sine  of  C  D  A  (=  D  A  B}  corresponds  to  the 
component  A  C ';  and  the  sine  of  C(=  sine  of  C  A  B)  corresponds- 
to  the  resultant  A  D.    Each  of  the  three  forces  is  therefore  repre- 
sented by  the  sine  of  the  angle  between  the  other  two. 

45.  Greatest  and  Least  Values  of  the  Resultant.— A 

change  in  the  angle  between  the  components  alters  the  value  of 
the  resultant ;  as  the  angle  increases  from  0°  to  180°,  the  resultant, 
diminishes  from  the  sum  of 

the  components  to  their  differ-  FlG-  9- 

ence.  In  Fig.  9,  let  C  A  B 
and  D  A  B  be  two  different 
angles  between  the  same  com- 
ponents A  C  (or  A  D}  and 
A  B.  As  C  A  £  is  less  than 
D  A  B,  its  supplement  A  B  F 
is  greater  than  ABE,  the 
supplement  of  D  A  B;  there- 
fore A  F  is  greater  than  A  E.  When  the  angle  C  A  B  is  dimin- 
ished to  0°,  the  sides  A  B,  B  F,  become  one  straight  line,  and  A  F 
equals  their  sum  ;  when  D  A  B  is  enlarged  to  180°,  E  falls  on 
A  B,  and  A  E  equals  the  difference  of  A  B  and  A  C.  Between 
the  sum  and  difference  of  the  comp'onents,  the  resultant  may 
have  all  possible  values. 

Two  forces  produce  equilibrium  when  they  are  equal  and  act 
upon  the  same  point  in  opposite  directions. 

Since  two  forces  produce  the  least  resultant  when  they  act  at 
an  angle  of  180°  with  each  other,  and  the  resultant  then  equals 
the  difference  of  the  forces,  if  the  forces  are  equal,  their  difference 


MECHANICS. 


FIG.  10. 


FIG.  11. 


is  zero,  and  the  resultant  vanishes ;  that  is,  the  two  forces  pro- 
duce equilibrium. 

46.  The  Polygon  of  Forces. — All  the  sides  of  a  polygon 
except  one  may  represent  so  many  forces  acting  at  the  same  time 
on  a  body,  and  the  remaining  side  will  represent  their  resultant. 
In  Fig.  10,  suppose  A  B,  A  C, 
and  A  D,  to  represent  three 
forces  acting  together  on  a  body 
at  A.  The  resultant  of  A  B  and 
A  C  is  represented  by  the  diag- 
onal A  E\  and  the  resultant 
ofAE  and  A  D  by  the  diagonal 
A  F.  As  A  F  is  equivalent  to 
A  E  and  A  D,  and  A  E  is  equiv- 
alent to  A  B  and  A  C}  therefore 
A  F  is  equivalent  to  the  three,  A  B,  A  C,  and  A  D.  But  if  we 
substitute  B  E  for  A  C,  and  E  F  for  A  D,  then  the  three  compo- 
nents are  A  B,  B  E,  and  E  F,  three  sides  of  a  polygon,  and  the 
resultant  A  F  is  the  fourth  side  of  the  same  polygon. 

So,  in  Fig.  11,  A  B,  B  C,  CD,  D  E,  and  E  F,  may  represent 
the  directions  and  relative  magni- 
tudes of  five  forces,  which  act  simul- 
taneously on  a  body  at  A.  The  re- 
sultant of  A  B  and  B  C  is  A  C ;  the 
resultant  of  A  C  and  C  D  is  A  D ; 
the  resultant  of  A  D  and  D  E  is 
A  E ;  and  the  resultant  of  A  E  and 
E  Fis  A  F;  which  last  is  therefore 
the  resultant  of  all  the  forces,  A  B, 
B  C,  C  D,  D  E,  and  E  F ;  the  com- 
ponents being  represented  by  five 
sides,  and  their  resultant  by  the  sixth  side,  of  a  polygon  of  six  sides. 

More  than  three  forces  in  one  plane  will  produce  equilibrium  when 
they  can  be  represented  by  the  sides  of  a  polygon  taken  in  order. 

Since  several  forces  acting  on  a  body,  are  represented  by  all 
the  sides    of    a  polygon    except 
one,  their  resultant  is  represented  FlG- 1^- 

by  the  remaining  side.    Thus,  the  _    ^       C 

resultant  of  the  forces  A  B,  B  C, 
C  D,  and  D  E  (Fig.  12),  is  A  E. 
Now,  the  force  E  A,  equal  and 
opposite  to  A  E,  since  it  would  be 
in  equilibrium  with  A  E,  is  there- 
fore in  equilibrium  with  all  the 


CURVILINEAR    MOTION. 


29 


F     13 


others.     Hence  the  forces  AS,  B  C,  C  D,  D  E,  and  E  A,  taken 
in  order  around  the  figure,  are  in  equilibrium. 

47.  Curvilinear  Motion.  —  Since,  according  to  the  first  law 
of  motion,  a  moving  body  proceeds  in  a  straight  line,  if  no  force 
disturbs  it,  whenever  we  find  a  body  describing  a  curve,  it  is  cer- 
tain that  some  force  is  continually  deflecting  it  from  a  straight 
line.      Besides  the   original    impulse,   therefore,  which  gave  it 
motion  in  one  direction,  it  is  subject  to  the  action  of  a  continued 
force,  which  operates  in  another  direction.     A  familiar  example 
occurs  in  the  path  of  a  projectile.     Suppose  a  body  to  be  thrown 
from  P  (Fig.  13),  with  an  impulse  which  would  alone  carry  it  to 
N,  in  the  same  time  in  which  gravity  alone  would  carry  it  to  V. 
Complete  the  parallelogram  P  Q  ; 

then,  as  both  motions  coexist  (2d 
law),  the  body  at  the  end  of  the 
time  will  be  found  at  Q.  Let  t  be 
the  time  of  describing  P  Nor  P  V; 
and  let  t'  be  the  time  of  describing 
P  M  by  the  impulse,  or  P  L  by 
gravity.  Then,  at  the  end  of  the 
time  t',  the  body  will  be  at  0.  Now, 
as  P  Nis  described  uniformly, 
PN-.PM::  t  :  *';.-.  P  N2  :  PJf2:: 
t2  :  f\ 

But  (Art.  25),  P  V:  P  L  ::  t2  : 
t'*;  .:  P  V:  PL  ::  P  N*:P  M*',  or  Q  V*  :  0 

Hence,  the  curve  is  such  that  P  Foe  Q  F2  ;  that  is,  the  abscissa 
varies  as  the  square  of  the  ordinate,  which  is  a  property  of  the 
parabola.  P  0  Q  is  therefore  a  parabola,  one  of  whose  diameters 

QV* 
is  P  V,  and  the  parameter  to  that  diameter  is  „  ^  • 

Owing  to  the  resistance  of  the  air,  the  curve  deviates  sensibly 
from  a  parabola,  especially  in  swift  motions. 

48.  Calculation  of  the  Resultant  of  Two  Impulsive 
Forces.  —  When  two  components  and  the  angle  between  them  are 
given,  the  resultant  may  be  found  both  in  direction  and  magni- 
tude by  trigonometry.     The  theorem  required  is  that  for  solving  a 
triangle,  when  two  sides  and  the  included  angle  are  given  ;  but 
the  included  angle  is  not  that  between  the  components,  but  its 
supplement  (Art.  43).    In  Fig.  7,  if  A  B  —  54,  and  A  C  =  22,  and 
C  A  B  =  75°,  then  A  CD  is  the  triangle  for  solution,  in  which 
A   C  =  22,   C  D  =  54,  and  A    C  D  =  105°.     Performing  the 
calculation,  we-find  the  resultant  A  D  =  63.363,  and  the  angle 


30 


MECHANICS. 


DAB,  which  it  makes  with  the  greater  force,  =  19°  35'  43", 
This  method  will  apply  in  all  cases. 

1.  A  foot-ball  received  two  blows  at  the  same  instant,  one 
directly  east  at  the  rate  of  71  feet  per  second,  the  other  exactly 
northwest,  at  the  rate  of  48  feet  per  second ;  in  what  direction 
and  with  what  velocity  did  it  move  ? 

Ans.  N.  47°  30'  52"  E.    Vel.  =50.253. 

The  process  is  of  course  abridged,  if  the  forces  act  at  a  right 
angle  with  each  other,  as  in  the  following  example  : 

2.  A  balloon  rises  1120  feet  in  one  minute,  and  in  the  same 
time  is  borne  by  the  wind  370  feet ;    what  angle  does  its  path 
make  with  the  vertical,  and  what  is  its  velocity  per  second  ? 

Ans.  18°  16'  53";  v  =  19.659. 

In  the  next  example,  one  component  and  the  angle  which  each 
component  makes  with  the  resultant,  are  given  to  find  the  result- 
ant and  the  other  component. 

3.  From  an  island  in  the  Straits  of  Sunda,  we  sailed  S.  E.  by 
S.  (33°  45')  at  the  rate  of  6  miles  an  hour ;  and  being  carried  by  a 
current,  which  was  running  toward  the  S.  W.  (making  an  angle 
with  the  meridian  of  64°  12£'),  at  the  end  of  four  hours  we  came 
to  anchor  on  the  coast  of  Java,  and  found  the  said  island  bearing 
due  north ;  required  the  length  of  the  line  actually  described  by 
the  ship,  and  the  velocity  of  the  current  ? 

Ans.  s  =  26.4  miles. 

v  ==  3. 7024  miles  per  hour. 

If  the  magnitudes  and  directions  of  any  number  of  forces  are 
given,  the  resultant  of  them  all  is  obtained  by  a  repetition  of  the 
same  process  as  for  two.  In  Fig.  11,  first  calculate  A  C,  and  th& 
angle  A  C  B,  by  means  of  A  B,  B  C,  and  the  angle  B.  Subtracting" 
A  C B  from  B  CD,  we  have  the  same  data  in  the  next  triangle, 
to  calculate  A  D,  and  thus  proceed  to 
the  final  resultant,  A  F. 

As  it  is  immaterial  in  what  order 
the  components  are  introduced  into 
the  calculation,  it  will  diminish  labor, 
to  find  first  the  resultant  of  any  two 
equal  components,  or  any  two  which 
make  a  right  angle  with  each  other ; 
since  it  can  be  done  by  the  solution 
of  an  isosceles,  or  a  right-angled  tri- 
angle. 

4.  The    particle    A    (Fig.    14)     is 
urged  by    three    equal    forces    A    B, 

A  C,  and  A  D  j    the  angle  B  A  0=  90°,  and  G  A  D  =  45C 


FIG.  14. 


JL* 


RESULTANT  OP  IMPULSIVE   FORCES. 


31 


what  is  the   direction   of   the   resultant,   and   how  many   times 
A  B  ?  Ans.  B  A  F  =  80°  16',  and 

A  F  :A  B  ::  ^/3:l. 

5.  Five  sailors  raise  a  weight  by  means  of  five  separate  ropes, 
in  the  same  plane,  connected  with  the  main  rope  that  is  fastened 
to  the  weight  in  the  manner  represented  in  Fig.  16.  B  pulls  at 
an  angle  with  A  of  20°;  (7  with  B,  19°;  D  with  C,  21°  30';  and 
E  with  Z>,  25°.  A,  B,  and  C,  pull  with  equal  forces,  and  D  and 
E  with  forces  one-half  greater  ;  required  the  magnitude  and 
direction  of  the  resultant. 

Ans.  Its  angle  with.  A  is  46°  33'  10".     Its  magnitude  is  5.1957 
times  the  force  of  A. 


FIG.  16. 


FIG. 


If  the  polygon  0  A  B  C  D  E  (Fig.  15)  be  constructed  for  the 
above  case,  0  B  and  C  E  are  easily  calculated  in  the  isosceles  tri- 
angles 0  A  B  and  ODE,  after  which  0  C  and  then  0  E  are  to 
be  obtained  by  the  general  theorem. 

49.  The  Resultant  and  all 
Components,  except  one,  being 
given,  to  Find  that  one  Compo- 
nent.— If  A  B  (Fig.  17)  is  the  result- 
ant to  be  produced,  and  there  already 
exists  the  force  A  C,  a  second  force 
can  be  found,  which  acting  jointly  with 
A  C,  will  produce  the  motion  required. 
Join  C  B,  and  draw  A  D  equal  and  par- 
allel  to  it,  then  A  D  is  the  force  re- 


32  MECHANICS. 

quired  ;  f or  A  B  is  equivalent  to  A  C  and  C  B.  Therefore  C  E- 
has  the  magnitude  and  direction  of  the  required  force ;  AD  is 
the  line  in  which  it  must  act. 

Again,  suppose  that  several  forces  act  on  A,  and  it  is  required 
to  find  the  force  which,  in  conjunction  with  them  all,  shall  pro- 
duce the  resultant  A  B.  Let  the  several  forces  be  combined  into 
one  resultant,  and  let  A  C  represent  that  resultant.  Then  A  1) 
may  be  found  as  before. 

The  trigonometrical  process  for  finding  a  component  is  essen- 
tially the  same  as  for  finding  a  resultant. 

1.  A  ferry-boat  crosses  a  river  £  of  a  mile  broad  in  45  minutes, 
the  current  running  all  the  way  at  the  rate  of  3  miles  an  hour  ;  at 
what  angle  with  the  direct  course  must  the  boat  head  up  the  stream 
in  order  to  move  perpendicularly  across?  Am.  71°  34'. 

2.  A  sloop  is  bound  from  the  mainland  of  Africa  to  an  island 
bearing  "W.  by  N.  (78°  45')  distant  76  miles,  a  current  setting 
N.  N.  W.  (22°  30')  3  miles  an  hour ;  what  is  the  course  to  arrive 
at  the  island  in  the  shortest  time,  supposing  the  sloop  to  sail  at 
the  rate  of  6  miles  per  hour  ;  and  what  time  will  she  take  ? 

Ans.  Course,  S.  76°  41'  4"  W.     Time,  10  h.  40  m.  7  sec. 

3.  The  resultant  of  two  forces  is  10  ;  one  of  them  is  8,  and  the 
direction  of  the  other  is  inclined  to  the  resultant  at  an  angle  of  36°. 
Find  th'e  angle  between  the  two  forces. 

Ans.  47°  17'  5"  or  132°  42'  55". 

4.  A  ball  receives  two  impulses  :  one  of  which  would  carry  it 
N.  27  feet  per  second  ;  the  other  K  60°  E.  with  the  same  velocity  ; 
what  third  impulse  must  be  conjoined  with  them,  to  make  the  ball 
go  E.  with  a  velocity  of  21  ft  ?     Ans.  S.  3°  22'  W.    v  =  40.57. 

50.  Resolution  of  Motion. — In  the  composition  of  motions 
or  forces,  the  resultant  of  any  given  components  is  found  ;  in  the 
resolution  of  motion  or  force,  the  process  is  reversed ;  the  resultant 
being  given,  the  components  are  found,  which  are  equivalent  to- 
that  resultant.  t 

If  it  be  required  to  find  what  FIG.  18. 

two  components  can  produce  the 
resultant  A  B  (Fig.  18),  we  have 
only  to  construct  on  A  B,  as  a  base, 
any  triangle  whatever,  as  A  B  C  or 
A  BD  (Art.  43);  then,  if  A  C  is 
one  component,  the  other  is  A  F, 
equal  and  parallel  to  G  B ;  or  if 
A  D  is  one,  the  other  is  A  E,  equal 
and  parallel  to  D  B  ;  and  so  for 
any  triangle  whatever  on  the  base  A  B.  The  number  of  pairs  is 
therefore  infinite,  whose  resultant  in  each  case  is  A  B. 


RESOLUTION    OF    A    FORCE.  33 

The  directions  of  the  components  may  be  chosen  at  pleasure, 
provided  the  sum  of  the  angles  made  with  A  B  is  less  than  two- 
right  angles. 

The  magnitude  and  direction  of  one  component  may  be  fixed 
at  pleasure. 

The  magnitudes  of  both  components  may  be  what  we  please, 
provided  their  difference  is  not  greater,  and  their  sum  not  less, 
than  the  given  resultant. 

These  conditions  are  obvious  from  the  properties  of  the 
triangle. 

When  a  given  force  has  been  resolved  into  two  others,  each, 
of  those  may  again  be  resolved  into  two,  each  of  those  into  two 
others  still,  and  so  on.  Hence  it  appears  that  a  given  force  may 
be  resolved  into  any  number  of  components  whatever,  with  such 
limitations  as  to  direction  and  magnitude  as  accord  with  the  fore- 
going statements. 

1.  A  motion  of  153  toward  the  north  is  produced  by  the  forces- 
100  and  125  ;  how  are  they  inclined  to  the  meridian  ? 

Ans.  54°  28'  and  40°  37'  7". 

2.  A  resultant  of  617  divides  the  angle  between  its  components 
into  28°  and  74°  ;  what  are  the  components? 

Ans.  606.34  and  296.14. 

61.  Resolution  of  a  Force,  to  Find  its  Efficiency  in 
a  Given  Direction. — By  the  resolution  of  a  force  into  two 
others  acting  at  right  angles  with  each  other,  it  is  ascertained 
how  much  efficiency  it  exerts  to  produce  motion  in  any  given 
direction.  For  example,  a  weight  W  (Fig.  19),  lying  on  a  hori- 

FIG,  19. 


zontal  plane,  and  pulled  by  the  oblique  force  C  A,  is  prevented 
by  gravity  from  moving  in  the  line  C  A,  and  is  compelled  to 
remain  on  the  plane.  Resolve  C  A  into  C  B,  in  the  plane,  and 
C  D  perpendicular  to  it:  then  the  former  represents  the  compo- 
nent which  is  efficient  to  cause  motion  along  the  plane  ;  the  latter 
has  no  influence  to  aid  or  hinder  that  motion  ;  it  simply  dimin- 
ishes pressure  upon  the  plane.  In  like  manner,  if  A  C  is  an 
oblique  force,  pushing  the  weight,  its  horizontal  component,  B  C, 
3 


34: 


MECHANICS. 


is  alone  efficient  to  move  it  ;  the  other,  A  B,  merely  increasing 
the  pressure.  Representing  by  F  the  whole  force,  by  /  and  /'  the 
-components  in  direction  of  motion  and  resistance  respectively,  and 
by  a  the  angle  of  inclination,  we  have 

/  =  Fcosa 
and    f  =  F  sin  a. 

If  only  88  per  cent,  of  the  strength  of  a  horse  is  efficient  in 
moving  a  boat  along  a  canal,  what  angle  does  the  rope  make  with 
the  line  of  the  tow-path?  Ana.  28°  21'  27". 

52.  Resultant  found  by  means  of  Rectangular  Axes.  — 

When  several  forces  act  in  one  plane  upon  a  body,  their  resultant 
may  be  conveniently  found  by  the  use  of  right-angled  triangles 
alone.  Select  at  pleasure  two  lines  at  right  angles  to  each  other, 
both  of  them  lying  in  the  plane  of  the  forces,  and  passing  through 
the  point  at  which  the  forces  are  applied.  These  lines  are  called 
•axes.  The  following  example  illustrates  their  use  : 

Let  P  A,  P  B,  P  0,  P  D,  P  E  (Fig.  20)  represent  the  forces 
in  Question  5  (Art.  48).  Let  one  axis,  for 
convenience,  be  chosen  in  the  direction  P  A, 
and  let  P  H  be  drawn  at  right  angles  to  it 
for  the  other  axis.  These  axes  are  supposed 
to  be  of  indefinite  length.  Then  proceed  as 
in  Art.  51  to  resolve  each  force  into  two 
•components  on  these  axes.  As  P  A  acts  in 
the  dix-ection  of  one  axis,  it  does  not  need  to 
be  resolved.  The  Projections  of  P  B  are 

Pb   =  P  B  x  cos  20° 

P  b'  =  P  B  x  sin  20°; 


FIG.  20. 

C  (I 


again, 


PC   =  P  C  x  cos  39° 

PC'  =  P  G  x  sin  39°,  etc. 

Suppose  P  A  produced  so  as  to  equal  P  A  +  JPb  +  P  c  + 
JPd  +  Pe  —  M,  and  P  H  produced  so  as  to  equal  P  b'  •+•  PC'  + 
fd'-\-Pe'  =  N.  Now,  as  M  acts  in  the  line  PA,  and  N  at 
right  angles  to  it,  their  resultant  and  the  ahgle  which  it  makes 
with  P  A  are  found  by  the  solution  of  another  right-angled 
triangle.  The  resultant  is  5.1957,  and  the  angle  is  46°  33'  10",  as 
in  Art.  48. 

If  any  components  of  the  resolved  forces  are  opposite  to  P  A 
•or  P  H,  they  are  reckoned  as  negative  quantities. 

53.  Analytical  Expression  for  the  Resultant. — Put  A  C 

{Fig.  21)    =  P,  A  B  =  P',  A  D  =  R,  angle  C  A  B  =  a  ;  then  in 
triangle  A  B  D  we  have,  by  Geometry,   A  D*  =  "AW  + 


PRINCIPLE    OF    MOMENTS. 


35 


A  B  x  BE,  but  B E  =  B D  cos  a  =  P  cos  a,  and  hence  sub- 


stituting as  above  R*  =  P'2  +  P2  + 
2  P'  P  cos  a  ;  whence 


FIG.  21. 


J2  =  VP'2  +  P2  +  2  P'  P  cos  a  .  (1.) 

Hence,    7%e   resultant   of   any   two 

forces,  acting  at  the  same  point,  is 

equal  to  the  square  root  of  the  sum 

of  the  squares  of  the  two  forces,  plus  twice  the  product  of  the  forces 

into  the  cosine  of  the  included  angle. 

If  a  =  0,  its  cosine  will  be  1,  and  (1)  becomes 

R  =  P  +  P'. 
If  a  —  90°,  its  cosine  will  be  0,  and  we  shall  have 

R  =  VP*  +  P'*. 

If  a  =  180°,  its  cosine  will  be  —  1,  and  we  shall  have 
R  =  P  —  P'. 

1.  Two  forces,  P  and  P',  are  equal  in  intensity  to  24  and  30, 
respectively,  and  the  angle  between  them  is  105°  ;  what  is  the 
intensity  of  their  resultant  ?  Ans.  33.21. 

2.  Two  forces,  P  and  P',  whose  intensities  are,  respectively, 
equal  to  5  and  12,  have  a  resultant  whose  intensity  is  13  ;  required 
the  angle  between  them.  An.  90°. 

3.  A  boat  is  impelled  by  the  current  at  the  rate  of  4  miles  per 
hour,  and  by  the  wind  at  the  rate  of  7  miles  per  hour  ;  what  will 
be  her  rate  per  hour  when  the  direction  of  the  wind  makes  an 
angle  of  45°  with  that  of  the  current?  Ans.  10.2  miles. 

4.  Two  forces  and  their  resultant  are  all  equal ;  what  is  the 
value  of  the  angle  between  the  two  forces  ?  Ans.  120°. 

54.  Principle  of  Moments. — The  moment  of  a  force,  with 
respect  to  a  point,  is  the  product 
of  the  force  into  the  perpendicu- 
lar let  fall  from  the  point  to  the 
line  of  direction  of  the  force. 

The  fixed  point  is  called  the 
centre  of  moments ;  the  perpen- 
dicular distance,  the  lever-arm 
of  the  force;  and  the  moment 
measures  the  tendency  of  the 
force  to  produce  rotation  about 
the  centre  of  moments. 

Denote  the  forces  (Fig.  22) 
by  P,  P'  and  their  resultant  by 
R.  From  E  any  point  in  the 


FIG.  22. 


36  MECHANICS. 

plane  of  the  forces  let  fall  upon  the  directions  of  the  forces 
the  perpendiculars  E  a,  EC,  Ed.  Kepresent  these  by  I,  L,  I'. 
Draw  two  rectangular  axes  of  reference  as  in  Art.  52,  so  that 
one  of  them  may  pass  through  A  and  E.  The  projection  of  the 
resultant  R  is  equal  to  the  sum  of  the  projections  of  its  compo- 
nents (Art.  52) ;  hence, 

AY  =  AY"  +  AY' (1), 

By  similar  triangles  Ac  E  and  A  R  Y,  we  have 

AY:  EC=L  ::  AR  =  R  :  AE    .:  A  Y=  ^-X 
by  similar  triangles  A  a  E  and  A  P  Y",  we  have 

A  Y"  :Ea  =  l  ::  A  P  =  P  :  AE    .'.  A  Y"  = 
and  by  similar  triangles  Ad  E  and  A  P'  Y',  we  have 

A  Y'  :  Ed  =  I'  : :  A  P'  =  P'  :  A  E     .-.  A  Y1  =  l  *  _f  ; 

A   JLJ 

substituting  these  values  in  Eq.  (1)  and  multiplying  by  A  E,  we- 
have 

R  x  L  =  P  x  2  +  P'  x  I'. 

Hence,  the  moment  of  the  resultant  of  two  forces  with  respect  to- 
any  point  is  equal  to  the  algebraic  sum  of  the  moments  of  the  forces 
taken  separately. 

By  using  the  resultant  as  above  and  'a  third  force  the  moment 
of  the  resultant  of  the  three  forces  may  be  proved  equal  to  the 
algebraic  sum  of  the  moments  of  the  forces,  and  so  oh  for  any 
number  of  forces. 

To  illustrate  the  application  of  the  principle  of  moments,  sup- 
pose a  weight  W  of  1000  Ibs.  to  be  suspended  from  the  end  of  a 
spar  D  E  QS,  in  the  figure  ; 

required  the  strain  upon  the  v. 

stay  E  B. 

We  have  three  forces  in 
equilibrium  acting  at  E,  viz., 
the  weight  W,  the  strain  upon 
the  rope  S,  and  the  upward 
thrust  of  the  spar  T.  If  we 

select  the  centre  of  moments  x  /  \    •     | 

upon  the  line  of  either  force, 
the  moment  of  that  force  will 
be  zero.  As  the  thrust  T  is  equal  and  opposed  to  the  resultant 
of  S  and  W,  we  will  take  D  as  the  centre  of  moments,  and  we 
have, 

Moment  of  T  =  moment  01*  S  +  moment  of  W ;  but  T  x  0  = 
moment  of  T,  S  x  KD  =  moment  of  S,  and  W  x  D  n  =  mo- 


PARALLEL    FORCES.  3f 

ment  of  W.  But  W  tends  to  cause  E  to  revolve  towards  the  right 
about  the  poin  t  D,  while  S  tends  to  cause  revolution  of  E  towards 
the  left  ;  hence  one  must  be  regarded  as  a  positive  and  the  other 
as  a  negative  moment,  and  we  have,  finally,  if  we  call  W  x  D  n 

positive,  0  =  -  8  x  KD  +  W  x  n  D,  whence,  S  =  ?*  j^^- 

If  in  the  problem,  D  E  =  20  ft,  B  D  =  20  ft.,  and  E  B  D  =  30°, 
then  will  KD  =  B  D  sine  30°  =  10,  and  Dn  =  D£sine  30°  =  10. 


To  find  the  thrust  at  D,   take  the  point  £  as  a  centre  of 
moments,  then 

T  x  Ba  =  S  X  0  +  W  x  Bn  ;  /.  T=  W  *  Bn  . 

-DO, 

Now  B  n  =  B  D  +  D  n  =  30  ;  to  find  B  a,  we  have 

K  E  —  ^DE*—inJz  =  V300,  and  B  E  =  2  K  E  =  2\/300l 
In  similar  triangles  K  D  E  and  B  E  a  we  have 

D  E:B  E  r.K  D  :  B  a  ; 
2  V300  x  10  1000  X30 

~~     =V30U;  hence  T^ 


Remember  that  when  three  forces,  acting  at  the  same  point, 
are  in  equilibrium,  one  of  the  three  being  known,  either  of  the 
other  two  can  be  found  by  taking  the  centre  of  moments  on  the 
line  of  the  force  not  sought,  and  equating  the  moments  of  the  two 
forces  considered. 

55.  Forces  Acting  at  Different  Points.  Parallel 
Forces. — We  have  thus  far  considered  forces  acting  upon  a 
single  particle,  or  upon  one 
point  of  a  body.  If,  how- 
ever, two  forces  P  and  P', 
in  the  same  plane,  act  upon 
A  and  B,  two  different 
points  of  a  rigid  body,  they 
may  still  have  a  resultant. 

Let  the  lines  of  direc- 
tions of  the  two  forces  A  F  and  B  D  (Fig.  24)  be  produced  to 
meet  in  C.  The  two  forces  may  then  be  considered  as  acting  at 
C,  and  thus  compounded  into  a  single  force  at  that  point,  or  at. 
the  point  G  of  the  body. 

Calling  the  angle  B  C  G=$  and  A  C  G=a  we  have,  projecting 
P'  and  P  upon  the  line  of  R, 

R  =  P'  cos  j3  +  P  cos  a  .  .  .  (1). 


38  MECHANICS. 

When  the  forces  become  parallel,  as  A  F  and  B  E,  ft  =  0,  and 
a  =  0,  and  (1)  becomes 

R  =  P'  +  P  .  .  .  (2). 

If  the  parallel  forces  act  in  opposite  directions,  as  A  F  and 
B  E',  then  a  =  180°,  and  0  =  0,  and  (1)  becomes 

R  =  P'  —  P .  .  .  (3).     Hence, 

The  resultant  of  two  parallel  forces  is  in  a  direction  parallel  to 
them  and  equal  to  their  algebraic  sum. 

56.  Point  of  Application  of  the  Resultant. — Let  P  and 
P'  (Figs.  25,  26)  be  two  parallel  forces  acting  in  the  same  or  in 

FIG.  25.  FIG.  26. 


fir 


opposite  directions,  and  let  E  be  the  point  of  application  of  the 
resultant.  Assume  this  point  as  a  centre  of  moments  ;  then  from 
Art.  54,  since  L  =  0, 

P  x  H  E  =  P1  x  G  E,  or,  in  the  form  of  a  proportion, 
P'  :  P  :  :  H  E  :  G  E.     But  by  similar  triangles, 
HE-.  QE'.'.AE-.E  B  ;  .: 
P'  :  P  :  :  A  E  :  E  B. 

That  is,  the  line  of  direction  of  the  resultant  of  two  parallel  forces 
divides  the  line  joining  the  points  of  application  of  the  components, 
inversely  as  the  components. 

By  composition  (Fig.  25)  and  division  (Fig.  26)  we  obtain 
P'  +  P  :  P  :  :  A  B  :  E  B,  and 
P  —  P'  :  P  :  :  A  B  :  E  B. 

That  is,  if  a  straight  line  be  drawn  to  meet  the  lines  of  two  parallel 
forces  and  their  resultant,  each  of  the  three  forces  will  be  propor- 
tional to  that  part  of  the  line  contained  between  the  other  two. 
"When  the  forces  act  in  the  same  direction,  we  have 

E  B  =  —p,  —  —-pi  and  when  they  act  in  opposite  directions, 
P  x  AB 


If,  in  the  last  case,  P  —  P',  then  E  B  will  be  infinite.  The 
two  forces  in  this  case  constitute  what  is  called  a  couple.  Their 
effect  is  to  produce  rotation  about  a  point  between  them. 

Any  number  of  parallel  forces  may  be  reduced  to  a  single  force 


EQUILIBRIUM  OP  FORCES.  39 

(or  to  a  couple)  by  first  finding  the  resultant  of  two  forces,  then 
the  resultant  of  that  and  a  third  force,  and  so  on  to  the  last.  And 
any  single  force  may  be  resolved  into  two  or  any  number  of  paral- 
lel forces  by  a  method  the  reverse  of  this. 

57.  Equilibrium  of  Parallel  Forces. — In  order  that  a  force 
may  be  in  equilibrium  with  two  parallel  forces, 

1.  It  must  be  parallel  to  them. 

2.  It  must  be  equal  to  their  algebraic  sum. 

3.  The  distances  of  its  line  of  action  from  the  lines  in  which  the 
two  forces  act,  must  be  inversely  as  the  forces. 

These  three  conditions  belong  to  the  resultant  of  two  parallel 
forces,  and  therefore  belong  'to  that  force  which  is  in  equilibrium 
with  the  resultant. 

68.  Equilibrium  of  Couples. — If  two  parallel  forces  are 
such  as  to  constitute  a  couple,  no  one  force  can  be  in  equilibrium 
with  them.  For  the  resultant  of  a  couple  is  zero,  and  has  its  point 
of  application  at  an  infinite  distance  (Art.  56).  But  a  couple  can 
be  held  in  equilibrium  by  another  couple  ;  and  the  second  couple 
may  be  either  larger  or  smaller  than  the  given  couple,  or  it  may 
be  equal  to  it. 

Let  the  couple  P  and  P'  (Fig.  27)  act  FIG.  27. 

on  a  body  at  the  points  A  and  B ;  they 
tend  to  produce  rotation  about  the  middle 
point  C.  If  another  couple,  Q  and  Q', 
equal  to  P  and  P',  should  be  applied  to 
produce  equilibrium,  one  must  directly 
oppose  P,  and  the  other  P'.  Then  A  and 
B,  being  each  held  at  rest,  all  the  forces 


P' 


are  in  equilibrium.  P 

But  if  the  second  couple  is  less  than 
P  and  P',  they  must  act  at  distances  from 
C,  which  are  as  much  greater  as  the  forces 
are  less  ;  or,  if  the  second  couple  is  greater 
than  the  first,  they  must  act  at  distances  — ; 

which  are  as  much  less.  Thus,  the  couple 
p  and  p,  acting  at  D  and  E,  tend  to  produce  rotation  about  C  in 
one  direction,  and  P  and  P'  in  the  opposite ;  and  these  tendencies 
are  equal  when  D  C  :  A  C ::  P  :p.  For,  since  the  opposite  forces, 
P  and  p,  are  inversely  as  their  distances  from  C,  their  resultant  is 
at  C,  and  is  equal  to  P  —  p  (Art.  55).  For  the  same  reason,  the 
resultant  of  P'  and  p'  is  at  C,  and  equal  to  P'  —  p'.  But  P — p  = 
P'  —  p',  and  they  act  in  opposite  directions.  Hence  C  is  at  rest, 
and  therefore  all  the  forces  are  in  equilibrium. 

59.  The  Parallelepiped  of  Forces.— Hitherto  forces  have 


40 


MECHANICS. 


FIG.  28. 


V 


been  considered  as  acting  in  the  same  plane.  But  if  forces  act  in 
different  planes,  the  solution  of  every  case  may  be  reduced  to  the 
following  principle,  called  the  parallelopiped  of  forces. 

Any  three  forces  acting  in  different  planes  upon  a  body  may  be 
represented  by  the  adjacent  edges  of  a  parallelopiped,  and  their  re- 
sultant by  the  diagonal  which  passes  through  the  intersection  of 
those  edges. 

Let  A  C,  A  D,  and  A  E  (Fig.  28),  be  three  forces  applied  in 
different  planes  to  the  body  at  A. 
Construct  the  parallelopiped  G  P, 
having  A  C,  A  D,  and  A  E,  for  its 
adjacent  edges,  and  from  A  draw  the 
diagonal  A  B.  The  section  through 
the  opposite  edges  A  C  and  P  B  is  a 
parallelogram,  and  therefore  A  B  is 
the  resultant  of  A  C  and  A  P,  and 
A  P  is  the  resultant  of  A  D  and  A  E.  SB  ? 

Hence  A  B  is  the  resultant  of  A  C,    ' 
A  D,  and  A  E. 

This  process  may  obviously  be  reversed,  and  a  given  force  may 
be  resolved  into  three  components  in  different  planes  along  the 
edges  of  a  parallelopiped,  having  such  inclinations  as  we  please. 

60.  Rectangular  Axes. — The  parallelopiped  generally 
chosen  is  that  whose  sides  are  rectangles ;  the  three  adjacent 
edges  of  such  a  solid  are  called  rectangular  axes.  All  the  forces 
which  can  possibly  act  on  a  body  may  be  resolved  into  equivalent 
forces  in  the  direction  of  three  such  axes.  And  since  all  forces 
which  act  in  tlie  direction  of  any  one  line  may  be  reduced  to  a 
single  force  by  taking  their  algebraic  sum,  therefore  any  number 
of  forces  acting  through  one  point  may  be  reduced  to  three  in  the 
direction  of  three  axes  chosen  at  pleasure. 

FIG.  29.  FIG.  30. 


Let  A  X,  A  Y  (Fig.  29)  be  at  right  angles  with  each  other, 
and  A  Z  perpendicular  to  the  plane  A  X  and  A  Y.     Let  A  B 


COMPONENTS    AND    RESULTANT.  41 

represent  a  force  acting  on  A.  Kesolve  A  B  into  A  C  on  the  axis 
A  Z,  and  A  P  in  the  plane  of  A  X,  A  Y ;  then  resolve  A  P  into 
A  D  and  A  E  on  the  other  two  axes.  Therefore,  A  C,  A  D,  and 
A  E  are  three  rectangular  forces,  whose  resultant  is  A  B. 

Let  the  axes  A  X,  A  Y,  A  Z,  be  produced  indefinitely  (Fig.  30) 
to  X',  Y',  Z' ;  then  their  planes  will  divide  the  angular  space 
about  A  into  eight  solid  right  angles,  namely :  A-X  Y  Z,  A-X  Y'  Zt 
A-X  Y'Z,  A-X'YZ,  above  the  plane  of  X  and  Y,  &nd  A-X  Y  Z1, 
A-X  Y'  Z,  A-X'  Y'  Z',  A-X  YZ  below  it. 


61.    Geometrical    Relation    of  Components    and    Re- 


sultant.— A  force  acting  on  the  body 
A  may  be  situated  in  any  one  of  the 
eight  angles,  and  its  value  may  be 
expressed  in  terms  of  the  squares  of 
its  three  components.  Let  A  B 
(Fig.  31)  be  resolved  as  before  into 
the  rectangular  components  A  C, 
A  D,  and  A  E.  Then,  by  the  right- 
angled  triangles,  we  find 
A  B'  =  B  P*  +  A  Pa  =  A  I?  +  A  P1 ;  JM 
and  if 

A  P  =  A  C?+CP'  =  AC*+AD>; 

.-.AB>  =  A  C*  +  A  D*  +  A 


FIG.  31. 


and     AB  =  V  A  G1  +  A  D*  +  A  E'. 

If  A  E  is  in  the  plane  of  X  and  Y,  the  component  on  the  axis 
of  Z  becomes  zero,  and  A  B  =  V  A  G1  +  A  D3,  and  similarly  for 
the  other  planes. 

62.  Trigonometrical  Relation  of  Components  and  Re- 
sultant.— Let  the  angles  which  A  B  makes  with  the  axes  of 
X,  Y,  Z,  respectively,  be  a,  ft,  y ;  that  is,  B  A  C  =  a,  B  A  D  =  ft 
B  A  E  =  y.  In  the  triangle  ABC,  right-angled  at  C,  we  have 

A  C  =  A  B  .  cos  a. 
In  like  manner, 

A  D  =  A  B  .  cos  /? ; 
and    A  E  =  A  B  .  cos  y. 

And  since  A  B  is  the  resultant  of  the  forces  A  C,  A  D,  and 
A  E,  it  is  the  resultant  of  A  B  .  cos  a,  A  B  .  cos  fi,  A  B  .  cos  y. 
In  general,  the  components  of  any  force  P,  when  resolved  upon 
three  rectangular  axes,  are  P  .  cos  a,  P .  cos  (3,  P  .  cos  y. 


42  MECHANICS. 

63.  Any  Number  of  Forces  Reduced  to  Three  on 
Three  Rectangular  Axes. — Suppose  the  body  at  A  to  be  acted 
upon  by  a  second  force  P ' ,  whose  direction  makes  with  the  axes 
the  angles  a ',  j3',  y'  ;  then,  as  before,  P'  is  the  resultant  of  P' .  cos  a', 
P' .  cos  0',  P' .  cos  y' ;  and  a  third  force  P",  in  like  manner,  has 
for  its  components  P"  .  cos  a",  P"  .  cos  j3",  P"  .  cos  y"  ;  and  so  of 
any  number  of  forces. 

Now,  all  the  components  on  one  axis  may  be  reduced  to  one 
force  by  adding  them  together.  Hence,  the  whole  force  in  the 
axis  of  X  =  P .  cos  a  +  P' .  cos  a'  +  P" .  cos  a"  -f  P'" .  cos  a'"  +  &c.  ^ 
the  whole  in  the  axis  of  Y, 

=  P .  cos  /J  +  P' .  cos  0'  -f  P" .  cos  0"  +  P'" .  cos  0"'  +  &c. ; 
and  that  in  the  axis  of  Z, 

=  P.  cos  y  +  P' .  cos  y'  +  P".  cos  y"  +  P'" .  cos  y'"  +  &c. 

If  any  component  acts  in  a  direction  opposite  to  others  in  the 
same  axis,  it  is  affected  by  a  contrary  sign,  so  that  the  force  in  the 
direction  of  any  axis  is  the  algebraic  sum  of  all  the  individual 
forces  in  that  axis. 

If  the  sum  of  the  components  in  one  axis  is  reduced  to  zero  by 
contrary  signs,  the  effect  of  all  the  forces  is  limited  to  the  plane 
of  the  other  axes,  and  is  to  be  obtained  as  in  Art.  52,  where  two 
axes  were  employed.  If  the  sum  of  the  components  on  each  of 
two  axes  is  reduced  to  zero,  then  the  whole  force  is  exerted  in  the 
direction  of  the  remaining  axis,  and  is  therefore  perpendicular  to 
the  plane  of  the  other  two. 

64.  Equilibrium  of  Forces  in  Different  Planes. — Since 
all  the  forces  which  can  operate  on  a  body  may  be  reduced  to 
three  forces  on  rectangular  axes,  it  is  obvious  that  the  whole  sys- 
tem of  forces  cannot  be  in  equilibrium  till  the  sum  of  the  compo- 
nents on  each  axis  is  reduced  to  zero.  We  must  have,  therefore, 
in  Art.  63,  as  conditions  of  equilibrium,  these  three  equations  for 
the  three  axes,  X,  Y,Z\ 

P .  cos  a  +  P' .  cos  a'  +  P"  .  cos  a"  +,  &c.,  =  0  ; 
P .  cos  P  +  P' .  cos  0'  +  P"  .  cos  0"  +  ,  &c.,  =  0  ; 
P .  cos  y  +  P' .  cos  y'  +  P"  .  cos  y"  +,  &c.,  =  0. 

65.  Forces  Resisted  by  a  Smooth  Surface.— When- 
ever any  forces  cause  pressure  upon  a  surface  without  friction, 
and  are  held  in  equilibrium  by  its  resistance,  the  resultant  of 
those  forces  must  be  at  right  angles  to  the  surface.  Suppose  that 


FIG.  32. 


CENTRE    OF    GRAVITY.  43 

D  A  (Fig.  32)  is  either  a  single  force  or  the  resultant  of  two  or 
more  forces,  and  that  it  is  held  in  equili- 
brium by  the  reaction  of  A  B,  a  smooth  sur- 
face. If  D  A  is  not  perpendicular  to  the 
surface,  it  can  be  resolved  into  two  com- 
ponents, one  perpendicular  to  the  surface 
A  B,  the  other  parallel  to  it.  The  former, 
C  A,  is  neutralized  by  the  resistance  of 
the  surface  ;  the  latter,  B  A,  is  not  re- 
sisted, and  produces  motion  parallel  to 
the  surface,  contrary  to  the  supposition.  Therefore  D  A,  if  held 
in  equilibrium  by  the  surface  A  B,  must  be  perpendicular  to  it. 


CHAPTER   IY. 

THE    CENTRE    OF    GRAVITY. 

66.  The  Centre  of  Gravity  Defined.— Whenever  gravity 
pulls  a  body  towards  the  earth,  the  pull  is  the  resultant  of  the 
parallel  forces  exerted  by  the  earth  upon  the  separate  particles  of 
the  body.  Whatever  the  position  of  the  body  this  resultant  passes 
through  a  certain  point  called  the  centre  of  gravity.  Hence, 

The  centre  of  gravity  of  a  body  is  that  point  at  which  the  whole 
mass  of  the  body  may  be  considered  as  concentrated  ;  or, 

It  is  the  point  at  which  the  body,  if  supported  there,  and  if  acted 
upon  by  gravity  alone,  will  balance  in  every  position. 


67.  Centre  of  Gravity  of  Equal  Bodies  in  a  Straight 
Line. — The  centre  of  gravity  of  two  equal  particles  is  in  the  mid- 
dle point   between   them.     Let  A  and   B   (Fig.    33),    two   equal 
particles,  be  joined   by  a  straight  line,  and  let 
A  a  and  B  b  represent  the   forces   of  gravity. 
The  resultant  of  these  forces,    since  they  are 
parallel  and  equal,  will  pass  through  the   mid- 
dle of  A  B  (Art.  56) ;   G  is  therefore  the  centre  « 
of  gravity.     In  like  manner  it  is  proved  that 
the  centre  of   gravity  of  two   equal  bodies   is 
in   the  middle  point  between  their  respective  centres  of   gravity. 


44 


MECHANICS. 


FIG.  34. 


Any  number  of  equal  particles  or  bodies,  arranged  at  equal 
distances  on  a  straight  line,  have  their  common  centre  of  gravity 
in  the  middle ;  since  the  above  reasoning  applies  to  each  pair, 
taken  at  equal  distances  from  the  extremes.  Hence,  the  centre 
of  gravity  of  a  material  straight  line  (e.g.,  a  fine  straight  wire)  is 
in  the  middle  point  of  its  length. 

68.  Centre  of  Gravity  of  Regular  Figures. — In  the  dis- 
cussion of  the  centre  of  gravity  in  relation  to  form,  bodies  are  con- 
sidered uniformly  dense,  and  surfaces  are  regarded  as  thin  laminae 
-of  matter. 

In  plane  figures  the  centre  of  gravity  coincides  with  the  centre 
of  magnitude,  when  they  have  such  a  degree  of  regularity  that  there 
are  two  diameters,  each  of  which  divides  the  figure  into  equal  and 
symmetrical  parts. 

The  circle,  the  parallelogram,  the  regular  polygon,  and  the 
ellipse,  are  examples. 

For  instance,  the  regular  hexagon  (Fig.  34)  is  divided  sym- 
metrically by  A  B,  and  also  by  C  D.  Conceive 
the  figure  to  be  composed  of  material  lines 
parallel  to  A  B.  Each  of  these  has  its  centre 
of  gravity  in  its  middle  point,  that  is,  in  C  D, 
which  bisects  them  all  (Art.  67).  Hence,  the 
centre  of  gravity  of  the  whole  figure  is  in  CD. 
For  the  same  reason  it  is  in  A  B.  It  is,  there- 
fore, at  their  intersection,  which  is  also  the 
centre  of  magnitude. 

By  a  similar  course  of  reasoning  it  is  shown  that  in  solids  of 
uniform  density,  which  are  so  far  regular  that  they  can  be  divided 
symmetrically  by  three  different  planes,  the  centres  of  gravity  and 
magnitude  coincide  ;  e.g.,  the  sphere,  the  parallelepiped,  the 
cylinder,  the  regular  prism,  and  the  regular  polyhedron. 

69.  Centre  of  Gravity  between  Two  Unequal  Bodies. — 

The  centre  of  gravity  of  two  unequal  bodies  is  in  a  straight  line 

joining  their  respective  centres  of  gravity,  and  at  the  point  which 

divides  their  distance  in  the  inverse  ratio  of  their  weights.     Let 

A  a  and  B  b  (Fig.  35),  passing  through  the 

centres  of  gravity  of  A  and  B,  be  proportional 

to  their  weights,  and  therefore  represent  the 

forces  of  gravity  exerted  upon  them.    By  the 

laws  of  parallel  forces,  the  resultant  G  g  = 

Aa  +  B  b  (Art.  55),  and  A  a  :  B  b  : :  B  G : 

A  G.    Therefore  the  centre  of  gravity  must 

be  at  G,  through  which  the  resultant  passes 


FIG.  35. 


CENTRE    OF    GRAVITY. 


{Art.  66).     This  obviously  includes  the  case  of  equal  weights  (Art. 
67). 

It  appears  from  the  foregoing  that  the  whole  pressure  on  a 
support  at  G  is  A  +  B,  and  that  the  system  is  kept  in  equilibrium 
by  such  support. 

70.  Equal   Moments  with   Respect  to  the  Centre  of 
Gravity. — Applying  the  principle  FIG.  36. 

of  moments  we  have,  calling  the 
weights  W  and  W,  and  taking  the 
centre  of  moments  at  G  (Fig.  36) 

W  x  Gx=  W  x  Gy  ; 
but  G  x  :   Gy  ::  A  G  :  G  B ; 
.-.  W  x  A  G  =  W  x  Gil,  as  was 
proved  in  Art.  56. 

71.  Centre     of     Gr.avity    between    Three     or     More 
Bodies. — The  method  of  determining  the  centre  of  gravity  of 
two  bodies  may  be  extended  to  any  number. 

Let  A,  B,  G,  D,  &c.  (Fig  37),  be  the  weights  of  the  bodies, 


FIG.  37. 


and  let  the  centres  of  gravity  of  A  and  B  be 
connected  together  by  the  inflexible  line 
A  B. 

Divide  A  B  so  that  A  :  B  ::  B  G  :  A  G, 
or  A  +  B  :  B  ::  A  B  :  A  G;  then  G  is  the 
centre  of  gravity  of  A  and  B.  Join  G  G  ; 
and  since  A  +  B  may  be  considered  as  at  the  point  G,  divide 
G  G  so  that  A  +  B  +  C  :  G  ::  C  G  :  G  g.  In  like  manner,  K,  the 
centre  of  gravity  of  four  bodies,  is  found  by  the  proportion, 
A  +B  +  G  +D  :  D  ::  D  g  :  g  K.  The  same  plan  may  be  pursued 
for  any  number  of  bodies. 

72.  Centre  of  Gravity  of  a  Triangle. — The  centre  of  gravity 
of  a  triangle  is  one-third  of  the  distance  from  the  middle  of  a  side  on 
a  line  to  the  opposite  angle.     Bisect  A  C  in  D  (Fig.  38),  and  B  G  in 
E.    B  D  bisects  all  lines  across  the  triangle  parallel 

to  A  G ;  therefore  the  centre  of  gravity  of  all  those 
Jines — that  is,  of  the  triangle — is  in  B  D.  For 
a  like  reason,  it  is  in  A  E,  and  therefore  at  their 
intersection,  G.  Since  E  G  =  ^  B  C,  and  D  C 
=  \  A  G,  .'.  ED  =  %  A  B.  But  E  G  D  and 
A  G  B  are  similar  ;  .-.  D  G  :  B  G  ::  D  E  :  A  B  : :  A  D  c 

73.  Centre  of  Gravity  of  an  Irregular  Polygon. — Divide 

the  polygon  into  triangles  by  diagonals  drawn  through  one  of  its 


FIG.  38. 


4G 


MECHANICS. 


angles,  and  then  proceed  according  to  the  methods  already  given. 
Let  A  C  E  (Fig.  39)  be  an  irregular  polygon,  whose  centre  of  grav- 
ity is  to  be  found.  Divide 
it  into  the  triangles  P,  Q, 
R,  S,  by  diagonals  through 
A,  and  find  their  centres  of 

gravity  a,  b,  c,  d  (Art.  72).  XP/«\  "ftX       ^ — F 

Join  a  b,  and  divide  it  so 
that ab:aG::  P  +  Q:  Q; 
then  G  is  the  centre  of 
gravity  of  the  quadrilateral 
P  -f  Q.  Then  join  G  c,  and 


\  R.    By  proceeding  in  this 

manner  till  all  the  triangles 

are  used,  the  centre  of  gravity  of  the  polygon  is  found  at  the  last 

point  of  division. 

74.  Centre  of  Gravity  of  the  Perimeter  of  an  Irregu- 
lar Polygon. — Find  the  centre  of  gravity  of  each  side,  which  is 
at  its  middle  point,  and  then 

proceed  as  in  Art.  71,  the 

weight  of  each  line  being 

considered  proportional  to 

its  length.     Thus,  let  a,  b, 

c,   &c.,   be  the  centres  of 

gravity  of  the  sides,  A  B, 

B  C,  CD,   &c.  (Fig.  40); 

join  a  b,  and  divide  it  so 

that  ab:aG::AB  +  BO 

:  B  C ;  then  G  is  the  centre 

of  gravity  of  A  B  and  B  C. 

Next  join  G  c,  and  make  Gc:  Gg  : :  AB  +  BO  +  CD :  CD; 

then  g  is  the  centre  of  gravity  of  those  three  sides.     Proceed  in 

this  manner  till  all  the  sides  are  used. 

The  perimeter  of  a  polygon  having  the  degree  of  regularity  de- 
scribed in  Art.  68,  has  its  centre  of  gravity  at  the  centre  of  the 
figure,  as  may  be  easily  proved.  If  a  polygon  has  a  less  degree  of 
regularity  than  that,  the  centre  of  gravity  both  of  its  area  and  its 
perimeter  may  usually  be  found  by  methods  more  direct  and 
simple  than  those  given  for  polygons  wholly  irregular. 

75.  Centre  of  Gravity  of  a   Pyramid.— TJte  centre  of 
gravity  of  a  triangular  pyramid  is  in  the  line  joining  the  vertex 
and  the  centre  of  gravity  of  the  base,  at  one-fourth  of  the  distance 
from  the  base  to  the  vertex. 


CENTRE    OF    GRAVITY. 


Let  G  (Fig.  41)  be  the  centre  of  gravity  of  the  base  BDC\  and 
g  that  of  the  face  ABC.  The  line  A  G  passes  through  the  centre 
of  gravity  of  every  lamina  jrIG 

parallel  to  D  B  C,  on  account 
of  the  similarity  and  similar 
position  of  all  those  laminae ; 
.•.  the  centre  of  gravity  of  the 
pyramid  is  in  A  G.  For  a 
similar  reason,  it  is  in  D  g ; 
and  therefore  at  their  inter- 
section, 0.  Now EG=.\ED, 
and  Eg  •=.  \  E  A  ;  hence,  by 
similar  triangles,  g  G=%AD. 
But  Gg  0  and  A  0  D  are  also 
.similar ;  .-.  G  0  =  |  A  0  =  ±  A  G. 

From  this  it  is  readily  proved  that  the  centre  of  gravity  of 
every  pyramid  and  cone  is  one-fourth  of  the  distance  from  the 
centre  of  gravity  of  the  base  to  the  vertex. 

76.  Examples  on  the  Centre  of  Gravity. — 

1.  A,  B,  and  C  (Fig.  42),   weigh,  respectively,  3,  2,  and  1 
pounds,  A  B  =  5  ft.,  B  C  =  4  ft.,  and 

C  A  =  2  ft.      Find    the   distance   of 
their  centre  of  gravity  from  C. 

First,  from  the  given  sides  of  the 
triangle  A  B  C,  calculate  the  angles. 
A  is  found  to  bo  49°  27£'.  Next  find 
the  place  of  G,  the  centre  of  gravity  of 
A  and  B,  by  the  proportion,  A  +  B  :  B  ::  A  B  :  A  G ;  A  G  is 
2  ft.,  equal  to  A  C.  Calculate  C  G,  the  base  of  the  isosceles  tri- 
angle A  G  C.  Its  length  is  1.673.  Then  find  Gg  by  the  propor- 
tion C G  :  Cg  ::  A  +  B  +  C :  A  +  B]  therefore  Cg  =  1.394. 

2.  A  =  5  Ibs.,  B  =  3  Ibs.,  and   C  =  12  Ibs.;    A  B  =  8  ft., 
A  G  =  4  ft,  and  the  angle  A  is  90° ;  find  the  distance  of  the 
centre  of  gravity  of  A,  B,  and  C  from  C.  Ans.  2  ft. 

3.  Three  equal  bodies  are  placed  at  the  angles  of  any  triangle 
whatever  ;  show  that  the  common  centre  of  gravity  of  those  bodies 
•coincides  with  the  centre  of  gravity  of  the  triangle. 

4.  Find  the  centre  of  gravity  of  five  equal  heavy  particles 
placed  at  five  of  the  angular  points  of  a  regular  hexagon. 

Ans.  It  is  one-fifth  of  the  distance  from  the  centre  to 
the  third  particle. 

5.  A  regular  hexagon  is  bisected  by  a  line  joining  two  oppo- 
site angles  ;  where  is  the  centre  of  gravity  of  one-half  ? 

Ans.  Four-ninths  of  the  distance  from  the  centre  to  the 
middle  of  the  second  side. 


48  MECHANICS. 

6.  A  square  is  divided  by  its  diagonals  into  four  equal  parts, 
one  of  which  is  removed  ;  find  the  distance  from  the  opposite 
side  of  the  square  to  the  centre  of  gravity  of  the  remaining  figure. 

Ans.  T\-  of  the  side  of  the  square. 

7.  Two  isosceles  triangles  are  constructed  on  opposite  sides  of 
the  same  base,  the  altitude  of  the  greater  being  h,  and  of  the  less, 
h'  ;  where  is  the  centre  of  gravity  of  the  whole  figure  ? 

Ans.  On  the  altitude  of  the  greater  triangle,  at  a  distance 
from  the  common  base  equal  to  £  (h  —  h'). 

8.  The  base  and  the  place  of  the  centre  of  gravity  of  a  triangle 
being  given,  required  to  construct  the  triangle. 

9.  Given  the  base  and  altitude  of  a  triangle  ;  required  to  con- 
struct the  triangle,  when  its  centre  of  gravity  is  perpendicularly 
over  one  end  of  the  base. 

10.  On  a  cubical  block  stands  a  square  pyramid,  whose  base,, 
volume,  and  mass  are  respectively  equal  to  those  of  the  cube  ;. 
where  is  the  centre  of  gravity  of  the  figure  ? 

Ans.  One-eighth  of  the  height  of  the  c«be  above  its. 
upper  surface. 

77.  Centre  of  Gravity  of  Bodies  in  a  Straight  Line 
referred  to  a  Point  in  that  Line.  —  If  several  bodies  are  in  a 
straight  line,  their  common  centre  of  gravity  may  be.  referred  to  a 
point  in  that  line  ;  and  its  distance  from  that  point  is  obtained  by 
multiplying  each  weight  into  its  own  distance  from  the  same  point, 
and  dividing  the  sum  of  the  products  by  the  sum  of  the  weights. 
Let  A,  B,  G,  and  D,  represent  the  weights  of  several  bodies,  whose 
centres  of  gravity  are  in  the  straight  line  o  D  (Fig.  43).  Required. 

FIG.  43. 


Q 

the  distance  of  their  common  centre  of  gravity  from  any  point  o 
assumed  in  the  same  line.  Let  G  be  their  common  centre  of 
gravity  ;  then,  calling  R  the  resultant  of  the  several  weights  A,  B, 
C  and  D,  which  acts  at  the  point  G,  we  have  from  principle  of 
moments, 

R  x  oG  =  A  xAo+BxBo+Cx  C  o  +  D  x  D  o, 
and  since        R  =  A  +  B  +  C  +  D  (Art  55),  we  have 
r,_AxAo  +  BxBo+CxCo  +  DxDo 
A  +  B  +  C+D 

78.  Centre  of  Gravity  of  a  System  referred  to  a 
Plane.  —  If  the  bodies  are  not  in  a  straight  line,  they  may  be  re- 
ferred to  a  plane,  which  is  assumed  at  pleasure.  The  distance  of 


CENTRE    OF    GRAVITY. 


FIG.  44. 


«/ 


their  common  centre  of  gravity  from  that  plane  is  expressed  as 
before  :  multiply  each  weight  into  Us  own  distance  from  the  plane,. 
and  divide  the  sum  of  the  products  ~by  the  sum  of  the  bodies. 

Letp,  p'  p"  (Fig.  44),  represent  the  weights  of  several  bodies, 
whose  centres  of  gravity  are  at  those  points  respectively,  and  let 
A  C  be  the  plane  of  reference. 
Join  p  p',  and  let  g  be  the  com- 
mon centre  of  gravity  of  p  and 
p';  draw  p  x,  g  Tc,  p'  x'  at  right 
angles  to  the  plane  A  G,  and 
consequently  parallel  to  each 
other  ;  join  x  x',  and  since  the 
points  p,  g,  p',  are  in  a  straight 
line,  the  points  x,  k,  x'  will  also 
be  in  a  straight  line,  and  there- 
fore x  x'  will  pass  through  Jc. 
Join  gp",  and  let  G  be  the  com- 
mon centre  of  gravity  ofp,p',p"; 
draw  G  K,  p"  x",  perpendicular 
to  the  plane  ;  and  through  g 
draw  m  n  parallel  to  x  x'  meet- 
ing p  x  produced  in  n. 

Now  p  :  p'  ::  p'  g  :  p  g  ::  (by  sim.  triangles)  p'  m  :  p  n; 

.-.  p  x  p  n  =  p'  x  p'  m,  or  p  x(nx—px)  =  p'  x  (  p'  x'  —  mx')  ; 
but 

nx=gk=  m  x',  .:  p  x  (g  k  —p  x}  =  p1  x  (p1  x'  —  g  k}, 
and 


\ 


for  the  same  reason,  if  p  +  p'  is  placed  at  g,  we  have 

p"x"  _  p  xpx+p'  xp'x'  +p"  xp"x"  . 
~ 


(P  +P'}+P"~  P+p'+p" 

a  formula  which  is  applicable  to  any  number  of  bodies. 

Let  the  last  equation  be  multiplied  by  the  denominator  of  the 
fraction,  and  we  have 
(p  +  p'+p"+  &c.)  GI£=px  p%+p'  xp'  x'+p"  x  p"  z"  +  &c.: 

that  is,  the  moment  of  any  system  of  bodies  with  reference  to  a  given 
plane,  equals  the  sum  of  the  moments  of  all  the  parts  of  the  system 
with  reference  to  the  same  plane. 

79.  Centre  of  Gravity  of  a  Trapezoid.  —  As  an  example 
of  the  foregoing  principle,  let  it  be  proposed  to  find  the  centre  of 

4 


50  MECHANICS. 

gravity  of  a  trapezoid,  considered  as  composed  of  two  triangles. 

The  centre  of  gravity  of  the  trapezoid  A  C  (Fig.  45)  is  in  E  F, 

which  bisects  all  the  lines  of  the 

iigure  parallel  to  B  C.     Suppose 

G  to  be  the  centre  of  gravity  of 

the  trapezoid ;  through    G  draw 

K  M  perpendicular  to  the  bases. 

Let  KM  =  h,BC  =  B,AD  =  b, 

and  join  B  D. 

The  moment  of  the  trapezoid 
-with  reference  to  B  Cis 

(B+b)J^.GK. 

The  moment  of  the  upper  triangle  is  —  •  ^7i;  the  moment  of  the 

.    Bh    h 
lower  triangle  is  —  •  -  ; 

z.  h    ,  Bh    h       b  h    2  7 

/.  (B  +  b)  £  •  GK=  -=-  •  -  +  —  -  -  h  ;  whence 

GK=-B±^T'l'    ™GM=h-^fJ¥-'\  = 

r>  -\-  0       6  £>  -\-  0       6 

?*+**;    .:GM:GK::2B  +  b:B+2b. 

_/>    -|—    Q  Q 

By  similar  triangles 
GM:GK::EG:GF;  .:  E  G  :  G  F::  2  B  +  b  :  B  +  2b ',  or 

the  centre  of  gravity  of  a  trapezoid  is  on  the  line  which  bisects  the 
parallel  bases,  and  divides  it  in  the  ratio  of  twice  the  longer  plu* 
the  shorter  to  twice  the  shorter  plus  the  longer. 

1.  Four  bodies,  A,  B,  C,  D,  weighing,  respectively,  2,  3,  6, 
and  8  pounds  are  placed  with  their  centres  of  gravity  in  a  right 
line,  at  the  distance  of  3,  5,  7,  and  9  feet  from  a  given  point ; 
what  is  the  distance  of  their  common  centre  of  gravity  from  that 
given  point ;  and  between  which  two  of  the  bodies  does  it  lie  ? 

Ans.  Between  C  and  D ;  and  its  distance  from  the  given 
point  7-^  feet. 

2.  There  are  five  bodies,  weighing,  respectively,  1,  14,  21£,  22, 
•and  29£  pounds ;   a  plane  is  assumed  passing  through  the  last 
"body,  and  the  distances  of  the  other  four  from  the  plane  are,  re- 
spectively, 21,  5,  6,  and  10  feet ;  how  far  from  the  plane  is  the 
common  centre  of  gravity  of  the  five  bodies?  Ans.  5  feet. 

80.   Centrobaric    Mensuration. — The    properties  of   the 
centre  of  gravity  furnish  a  very  simple  method  of  measuring 


CENTROBARIC    MENSURATION.  5] 

surfaces  and  solids  of  revolution.  This  method  is  comprehended 
in  the  two  following  propositions,  known  as  the  theorems  of 
Ouldinus  : 

1.  If  any  line  revolve  about  a  fixed  axis,  wliicli  is  in  the  plane 
of  that  line,  the  SURFACE  which  it  generates  is  equal  to  the  product 
of  the  given  line  into  the  circumference  described  by  its  centre  of 
gravity. 

Let  any  line,  either  straight  or  curved,  revolve  about  a  fixed 
axis  which  is  in  the  plane  of  that  line  ;  and  let/,*/',/",/'",  etc., 
denote  elementary  portions  of  the  line,  d,  d',  d",  a"",  &c.,  the 
distances  of  these  portions,  respectively,  from  the  axis  ;  then  the 
surface  generated  By  /,  in  one  revolution,  will  be  2  n  df  ;  hence 
the  surface  generated  by  the  whole  line  will  be 

S  =  2  TT  (df  +  d'f  +  d"  f"  +  d'"f"  4-  &c.)  .  .  .  (1). 

Put  L  =  the  length  of  the  revolving  line,  and  G  =  the  dis- 
tance from  the  axis  to  the  centre  of  gravity  of  the  line  ;  then 
<Art.  78) 

G  L  =  df+d'f  +  d"f"  +  d"!f"  +  &c..  .  .  (2). 
Combining  (1)  and  (2),  we  have 

S=%7rGL    .........  (3). 

2.  If  a  plane  surface,  of  any  form  whatever,  revolve  about  a  fixed 
axis  which  is  in  its  own  plane,  the  VOLUME  generated  is  equal  to 
the  product  of  that  surface  into  the  circumference  described  by  its 
•centre  of  gravity. 

Let  any  plane  surface  revolve  about  an  axis  which  is  in  the 
plane  of  that  surface  ;  and  let/,/',  /",/"',  &c.,  denote  elementary 
portions  of  the  surface,  d,  d',  d",  d'",  &c.,  the  distances  of  these 
portions,  respectively,  from  the  axis  ;  then  the  volume  generated 
by  /  in  one  revolution  will  be  2  TT  df;  hence  the  volume  generated 
by  the  whole  surface  will  be 


'f'  +  d"f"  +  d'"f'"  +  &e.)  .  •  (4). 

Put  A  =  the  area  of  the  revolving  surface,  and  G  =  the  dis- 
tance from  the  axis  to  the  centre  of  gravity  of  that  surface  ;  then 
<Art.  78) 

AG  =  df+d'f'  +  d"f"  +  d"'f"  f  &c.,  .  .  .  (5). 
Substituting  in  (4),  we  have 

V=27rAG  .........   (6). 

As  an  illustration  of  the  first  theorem,  the  straight  line  C  D 
(Fig.  46),  revolving  about  the  center  C,  describes  a  circle  whose 


52  MECHANICS. 

surface  is  equal  to  C  D  into  the  circumference  of  the  circle  de- 
scribed by  its  centre  of  gravity,  E.    This  is  evident  also  from  the- 
consideration  that,  since  E  is  the  centre  of  the  line  C  D,  the  cir- 
cumference described  by  it  will  be  half  the 
length  of  the  circumference  A  D  B  ;  and  the  FIG.  46. 

area  of  a  circle  is  equal  to  the  product  of  the 
radius  into  half  the  circumference. 

The  second  theorem  is  illustrated  by  the 
volume  of  a  cylinder,  whose  height  =  h,  and 
the  radius  of  whose  base  =  r. 

Common  method  ;  base=rr  r2 ;  height =h'} 
.:  vol.  =  TT  r2  h. 

Centrobaric  method ;  revolving  area  =  r  h  ;  circumference 
described  by  the  centre  of  gravity  —  £  r  x  2  TT-  /.  vol.  =  r  h  . 
%  r  .  2  TT  =  TT  r*  li. 

81.  Examples. — 

1.  Suppose  the  small  circle  (Fig.  46)  to  be  placed  with  its 
plane  perpendicular  to  the  plane  of  the  paper,  and  revolved  about 
C,  the  point  D  describing  the  line  D  B  A  ;  required  the  content 
of  the  solid  ring.     If  C  D  =  R,  and  E  D  =  r,  then  the  area  re- 
volved =  TT  r2,  and  the  circumference  D  B  A -  =  2  TT  R  ;  /.  the  ring 
=  2  -n2  R  r2.     It  is  equal  to  a  cylinder  whose  base  is  the  circle 
ED,  and  whose  height  equals  the  line  D  B  A. 

2.  Find  the  convex  surface  of  a  cone  ;  slant  height  =  s  ;  and 
rad.  of  base  =  r.     The  line  revolved  being  s,  and  the  distance 
from  the  axis  to  its  centre  of  gravity,  \  r,  the  surface  is  TT  r  s. 

3.  A  square,  whose  side  is  one  foot,  is  revolved  about  an  axis 
which  passes  through  one  of  its  angles,  and  is  parallel  to  a  diago- 
nal ;  required  the  volume  of  the  figure  thus  formed. 

Ans.  TT  V%,  or  4.4429  cubic  ft. 

4.  Find  the  centre  of  gravity  of  a  semi-circumference.     In 
this  case   the  revolving   semi-circumference  ABC  (Fig.  47} 
generates  the  surface  of  a  sphere ;  hence,  taking  the 
diameter  A  C  as  an  axis,  calling  the  distance  of  the       FlG-  47- 
centre  of  gravity  G  from  the  axis  x  and  radius  r,  we 

have  .. 

4  TT  r2  =  2  IT  z  x  TT  r ;  /.  x  •= 

7T 

Hence  the  distance  of  the  centre  of  gravity  of  the  semi- 
circumference  from  the  centre  of  the  circle  is 

2  r 

—  =  .637  r. 

TT 

5.  Find  the  centre  of  gravity  of  a  semicircle.    The  revolving- 


DIFFERENT    KINDS    OF    EQUILIBRIUM.  53 

area  generates  a  sphere,  and  hence,  as  in  the  preceding  problem, 
we  have 

$nrs  =  2Trxx$irr*',   ,'.x  —  \~=  .424  r. 

O    7T 

In  any  case,  when  a  simple  expression  for  the  surface  generated 
by  a  revolving  line  can  be  found,  it  is  easy  to  find  the  centre  of 
gravity  of  the  line  by  this  method,  and  the  centre  of  gravity  of 
an  area  may  be  readily  found  from  the  expression  of  the  volume 
generated. 

82.  Support  of  a  Body. — A  body  cannot  rest  on  a  smooth 
plane  unless  it  is  horizontal ;  for  the  pressure  on  a  plane  (Art.  65) 
cannot  be  balanced  by  the  resistance  of  that  plane,  except  when 
perpendicular  to  it ;  therefore,  as  the  force  of  gravity  is  vertical, 
the  resisting  plane  must  be  horizontal. 

The  base  of  support  is  that  area  on  the  horizontal  plane  which 
is  comprehended  by  lines  joining  the  extreme  points  of  contact. 

If  there  are  three  points  of  contact,  the  base  is  a  triangle ;  if 
four,  a  quadrilateral,  &c. 

When  the  vertical  through  the  centre  of  gravity  (called  the 
line  of  direction)  falls  within  the  base,  the  body  is  supported  ;  if 
without,  it  is  not  supported.  In  the  body  A  (Fig.  48)  the  force 
of  gravity  acts  in  the  line 
G  F,  and  there  are  lines  of 
resistance  on  both  sides  of 
G  F,  as  G  C  and  G  E,  so 
that  the  body  cannot  turn 
on  the  edge  of  the  base, 
without  rising  in  an  arc 
whose  radius  is  G  C  or  G  E. 
But,  in  the  body  B,  there  is  resistance  only  on  one  side ;  and 
therefore,  if  the  force  of  gravity  be  resolved  on  G  C  and  a  perpen- 
dicular to  it,  the  body  is  not  prevented  from  moving  in  the  direc- 
tion of  the  latter,  that  is,  in  the  arc  whose  radius  is  G  C. 

If  the  line  of  direction  fall  at  the  edge  of  the  base,  the  least 
force  will  overturn  it. 

83.  Different  Kinds  of  Equilibrium.— If  the  base  is  re- 
duced to  a  line  or  point,  then,  though  there  may  be  support,  there 
is  no  firmness  of  support ;  the  body  will  be  moved  by  the  least 
force.     But  it  is  affected  very  differently  in  different  cases. 

When  it  is  moved  from  its  position  of  support  and  left,  it  will 
in  some  cases  return  to  it,  pass  by,  and  return  again,  and  continue 
thus  to  vibrate  till  it  settles  in  its  place  of  support  by  friction  and 
other  resistances.  This  condition  is  called  stable  equilibrium. 


MECHANICS. 


In  other  cases,  when  moved  from  its  position  of  support  and 
left,  it  will  depart  further  from  it,  and  never  recover  that  position 
again.  This  is  called  unstable  equilibrium. 

In  other  cases  still,  the  body,  when  moved  from  its  place  of 
support  and  left,  will  remain,  neither  returning  to  it  nor  depart- 
ing further  from  it.  This  is  called  neutral  equilibrium. 

84.  Stable  Equilibrium.— Let  the  body  (Fig.  49)  be  sus- 
pended on  the  pivot  A.    This  is  its  base  of  support.     While  the 
centre  of  gravity  is  below  A,  the  line  of  direction 

EOF  passes  through  the  base,  and  the  body  is 

supported.     Let  it  be  moved  aside,  and  the  centre 

of  gravity  be  left  at  G.     Let  G  R  represent  the 

force  of  gravity,  and  resolve  it  ftito  G  N  on  the 

line  A  G,  and  N  R,  or  G  B,  perpendicular  to  A  G. 

G  N  is  resisted  by  the  strength  of  A,  and  G  B 

moves   the  centre   of  gravity  in  the  arc  whose 

radius   is   A  G.     Hence  the    body  swings  with 

accelerated  motion  till  the  centre  of  gravity  reaches 

0,  where  the  force  G  B  becomes  zero.     But  by 

its  inertia,  the  body  passes  beyond  that  position, 

and  ascends  on  the  other  side,  till  the  retarding 

force  of  gravity  stops  it  at  g,  as  far  from  0  as  G  is. 

It  then  descends  again,  and  would  never  cease  to  oscillate  were 

there  no  obstructions. 

85.  Unstable  Equilibrium. — Next,  let  the  body  be  turned 
on  the  pivot  till  the  centre  of  gravity  G  is  at  P,  above  A  (Fig.  SO)* 
Then,  as  well  as  when  G  is  below  A,  the  body  is 
supported,  because  the  line   of  direction  E  P  F 

passes  through  the  base  A.  But  if  turned  and  left 
in  the  slightest  degree  out  of  that  position,  it  can- 
not recover  it  again,  but  will  depart  further  and 
further  from  it.  Let  G  R  represent  the  force  of 
gravity,  and  let  it  be  resolved  into  G  N,  acting 
through  A,  and  G  B  perpendicular  to  it.  The 
former  is  resisted  by  A  ;  the  latter  moves  G  away 
from  P,  the  place  of  support.  If  the  body  is  free 
to  revolve  about  A,  without  falling  from  it,  the 
centre  of  gravity  will,  by  friction  and  other  resist- 
ances, finally  settle  below  A,  as  in  the  case  of 
stable  equilibrium. 

86.  Neutral  Equilibrium. — Once  more,  suppose  the  pivot 
supporting  the  body  to  be  at  G,  the  centre  of  gravity ;  then,  in 


FIG.  50. 


CENTRE    OF    GRAVITY.  55 

whatever  situation  the  body  is  left,  the  line  of  direction  passes 
through  the  base,  and  the  body  rests  indifferently  in  any  position. 

These  three  kinds  of  equilibrium  may  be  illustrated  also  by 
bodies  resting  by  curved  surfaces  on  a  horizontal  plane.  Thus,  if 
a  cylinder  is  uniformly  dense,  it  will  always  have  a  neutral  equi- 
librium, remaining  wherever  it  is  placed.  But  if,  on  account  of 
unequal  density,  its  centre  of  gravity  is  not  in  the  axis,  then  its 
equilibrium  is  stable,  when  the  centre  of  gravity 
is  below  the  axis,  and  unstable  when  above  it. 

In  general,  there  is  stable  equilibrium  when 
the  centre  of  gravity,  on  being  disturbed  in  either 
direction,  begins  to  rise ;  unstable  when,  if  dis- 
turbed either  way,  it  begins  to  descend;  and 
neutral  when  the  disturbance  neither  raises  nor 
lowers  the  centre  of  gravity. 

87.  Questions  on  the  Centre  of  Grav- 
ity.- 

1.  A  frame  20  feet  high,  and  4  feet  in  diam- 
eter, is  racked  into  an  oblique  form  (Fig.  51), 
till  it  is  on  the  point  of  falling ;   what  is  its 
inclination  to  the  horizon  ?  Ans.  78°  27'  47". 

2.  A  stone  tower,  of  the  same  dimensions  as  the  former,  is  in- 
clined till  it  is  about  to  fall,  but  preserves  its  rectangular  form  ; 
what  is  its  inclination  ?  Ans.  78°  41'  24". 

3.  A  cube  of  uniform  density  lies  on  an  inclined  plane,  and  is 
prevented  by  friction  from  sliding  down  ;   to  what  inclination 
must  the  plane  be  tipped,  that  the  cube  may  just  begin  to  roll 
down  ?  Ans.  45°. 

4.  What  must  be  the  inclination  of  a  plane,  in  order  that  a 
regular  prism  of  any  given  number  of  sides  may  be  on  the  point 
of  rolling  down,  if  friction  prevents  sliding  ? 

Ans.  Equal  to  half  the  angle  at  the  centre  of  the  prism, 
subtended  by  one  side. 

5.  A  body  weighing  83  Ibs.  is  suspended,  and  drawn  aside  from 
the  vertical  9°  by  a  horizontal  force ;  what  pressure  is  there  on 
the  point  of  support,  and  what  force  urges  it  down  the  arc  ? 

Ans.  Pressiire  on  the  support,  84.03  Ibs. 
Moving  force,  12.984  Ibs. 

88.  Motion  of  the  Centre  of  Gravity  of  a  System  when 
one  of  the  Bodies  is  Moved. — 

When  one  body  of  a  system  is  moved,  the  centre  of  gravity  of 
the  system  nwves  in  a  similar  path,  and  its  velocity  is  to  that  of 


56  MECHANICS. 

the  moving  body  as  the  mass  of  that  body  is  to  the  mass  of  the  whole 
system. 

If  the  system  contains  but  two  bodies,  A  and  B  (Fig.  52),  sup- 
pose A  to  remain  at  rest,  while  B 
describes  the  straight  lines  B  C, 
C  D,  &c.,  the  centre  of  gravity  G 
will  in  the  same  time  describe  the 
similar  series,  G  H,  H  J,  &c. 
When  B  is  in  the  position  B,  and 
the  centre  of  gravity  at  G,  A  G  : 
A  B  :  :  B  :  A  +  B  ;  when  B  is 
at  C,  A  H  :  A  C  :  :  B  :  A  +  B ; 
.-.  A  G  :  A  B::AH:AC.  Hence 
G  H  .is  parallel  to  B  <?,  and  G  II :  B  C : :  B  :  A  +  B.  In  like 
manner,  H  J  :  C  D  : :  B  :  A  +  B,  &c.  Thus  all  the  parts  of  one 
path  are  parallel  to  the  corresponding  parts  of  the  other,  and  have 
a  constant  ratio  to  them.  Therefore  the  paths  are  similar.  As 
the  corresponding  parts  are  described  in  equal  times,  their  lengths 
are  as  the  velocities.  But  the  lengths  are  as  B  :  A  -f-  B ;  there- 
fore the  velocity  of  the  common  centre  of  gravity  is  to  that  of  the 
moving  body  as  the  mass  of  the  moving  body  is  to  the  mass  of  both. 
The  same  reasoning  is  applicable  when  the  body  moves  in  a  curve. 

If  the  system  contain  any  number  of  bodies,  and  the  centre  of 
gravity  of  the  whole  be  at  G,  then  the  centre  of  gravity  of  all 
except  B  must  be  in  the  line  B  G  beyond  G.  Suppose  it  to  be  at 
A,  and  to  remain  at  rest,  while  B  moves  ;  then  it  is  proved  in  the 
same  manner  as  before,  that  G,  the  centre  of  gravity  of  the  whole 
system,  moves  in  a  path  parallel  to  the  path  of  B,  and  with  a 
velocity  which  is  to  B's  velocity  as  the  mass  of  B  to  the  mass  of 
the  entire  system. 

89.  Motion  of  the  Centre  of  Gravity  of  a  System  when 
Several  of  the  Bodies  are  Moved, — 

When  any  or  all  of  the  bodies  of  a  system  are  moved,  the  centre 
of  gravity  moves  in  the  same  manner  as  if  all  the  system  were 
collected  there,  and  acted  on  by  the  forces  which  act  on  the  separate 
bodies. 

Let  A,  B,  C,  &c.  (Fig.  53),  belong  to  a  system  containing  any 
number  of  bodies,  and  let  M  be  the  mass  of  the  system.  Let  A 
be  moved  over  A  a,  B  over  B  b,  C  over  C  c,  &c.  And  first  sup- 
pose the  motions  to  be  made  in  equal  successive  times.  If  the 
centre  of  gravity  of  the  system  is  first  at  G,  then  that  of  all  the 
bodies  except  A  is  in  A  G  produced,  as  at  g.  While  A  moves  to  a, 
G  moves  in  a  parallel  line  to  H  (Art.  88),  and  G  H:  A  a::  A  :  M . 
In  like  manner,  when  B  describes  B  b,  the  centre  of  gravity  of  the 
other  bodies  being  at  ?i,  the  centre  of  gravity  of  the  system  de- 


MOTION   OF    CENTRE    OF    GRAVITY.  57 

scribes  the  parallel  line,  H  K,  and  H  K  :  B  &  : :  B  :  M ;  and  when 
C  moves,  K  L  :  C  c  : :  C :  M,  &c.  Now,  A  a  and  G  H  represent 
the  respective  velocities  of  the 
body  A,  and  the  system  M; 
therefore,  if  we  convert  the 
proportion  G  H  :  A  a  : :  A  :  M 
into  an  equation,  we  have  A  x 
A  a  =  M  x  GH;  that  is,  the 
momentum  of  the  body  A 
equals  the  momentum  of  the 
system  M.  It  therefore  re- 
quires the  same  force  to  move 

A  over  A  a  as  to  move  the  system  M  over  G  H.  The  same  is  true 
of  the  other  bodies.  If  then  the  several  forces  which  move  the 
bodies,  limiting  the  number  to  three,  for  the  present,  were  applied 
successively  to  the  system  collected  at  G,  they  would  move  it  over 
G  H,  HK,  K  L.  But  if  applied  at  once,  they  would  move  it  over 
fr  L,  the  remaining  side  of  the  polygon.  If,  therefore,  the  forces, 
instead  of  acting  successively  on  the  bodies,  were  to  move  A  over 
A  a,  B  over  B  b,  and  C  over  C  c,  at  the  same  time,  the  centre  of 
gravity  of  the  system  would  describe  G  L  in  the  same  time.  In  the 
same  way  it  may  be  proved,  that  whatever  forces  are  applied  to  the 
several  bodies  of  a  system,  the  centre  of  gravity  of  the  system  is 
moved  in  the  same  manner  as  a  body  equal  to  the  whole  system 
would  be  moved,  if  all  the  same  forces  were  applied  to  it. 

It  is  possible  that  the  centre  of  gravity  of  a  system  should 
remain  at  rest,  while  all  the  bodies  in  it  are  in  motion.  For,  sup- 
pose all  the  forces  acting  on  the  bodies  to  be  such  that  they  might 
be  represented  in  direction  and  intensity  by  all  the  sides  of  a  poly- 
gon, then,  since  a  single  body  acted  on  by  them  would  be  in  equi- 
librium, therefore  the  centre  of  gravity  of  the  system  would  remain 
at  rest,  though  the  bodies  composing  it  are  in  motion. 

90.  Mutual  Action  among  the  Bodies  of  a  System. — 

The  forces  which  have  been  supposed  to  act  on  the  several  bodies 
of  a  system  are  from  without,  and  not  forces  which  some  of  the 
bodies  within  the  system  exert  on  others.  If  the  bodies  of  a  sys- 
tem mutually  attract  or  repel  each  other,  such  action  cannot  affect 
the  centre  of  gravity  of  the  whole  system.  For  action  and  reac- 
tion are  always  opposite  and  equal.  Whatever  force  one  body 
exerts  on  any  other  to  move  it,  that  other  exerts  an  equal  force  on 
the  first,  and  the  two  actions  produce  equal  and  opposite  effects 
on  the  centre  of  gravity  between  them.  Therefore  the  centre  of 
gravity  of  a  system  remains  at  rest,  if  the  bodies  which  compose 
it  are  acted  on  only  by  their  mutual  attractions  or  repulsions. 

91.  Examples  on  the  Motion  of  the  Centre  of  Gravity.— 


58 


MECHANICS. 


FIG.  54. 


1.  Two  bodies,  A  and  B,  of  given  weights,  start  together  from 
D  (Fig.  54),  and  move  uniformly  with  given  velocities  in  the  direc- 
tions D  A  and  D  B ;  required  the  di- 
rection and  velocity  of  their  centre  of 

gravity. 

As  the  directions  of  D  A  and  D  B 
are  given,  we  know  the  angle  A  D  B  ; 
from  the  given  velocities,  we  also  know 

the  lines  D  A  and  D  B,  described  in  a        -^* — mjs 

certain  time.    Calculate  the  side  A  B,      /  \ 

and  the  angles  A  and  B.     Find  the 

place  of  the  centre  of  gravity  G  between  the  bodies  at  A  and  B- 
Then,  in  the  triangle  D  B  G,  D  B,  B  G,  and  angle  B  are  known, 
by  which  may  be  found  the  distance  D  G  passed  over  by  the 
centre  of  gravity  in  the  time,  and  B  D  G  the  angle  which  its  path 
makes  with  that  of  the  body  B. 

2.  Three  bodies  of  given  weight,  A,  B,  C,  in  the  same  time 
and  in  the  same  order,  describe  with  uniform  velocity  the  three 
sides  of  the   given    triangle 

ABC  (Fig.  55)  ;  required 
the  path  of  their  centre  of 
gravity. 

Let  G  be  their  centre  of 
gravity  before  they  move.  If 
they  move  successively,  G  de- 
scribes G  K,  K  L,  L  M,  par- 
allel to  the  sides  of  the  trian- 
gle, and  having  to  them  re- 
spectively the  same  ratios  as 


FIG.  55. 


B 


FIG.  56. 


the  corresponding  moving  bodies  have  to  the  sum  of  the  bodies 
(Art.  89).  Thus,  three  sides  of  the  polygon  are  known. ;  and  the- 
angle  K  =  B,  and  L  =  C.  These  data  are  sufficient  for  calcu- 
lating the  fourth  side,  G  M,  which  the  centre  of  gravity  describes,, 
when  the  bodies  move  together. 

3.  Show  that  when  the  three  bodies  in  Example  2  are  equal,, 
the  centre  of  gravity  will  remain  at  rest. 

4.  A  (Fig.  56)  weighs  one  pound  ; 
B  weighs  two  pounds,  and  lies  direct- 
ly east  of  A  ;    they  move  simulta- 
neously, A  northward,  and  B  east- 
ward, at  the  same  uniform  rate  of  40 
feet  per  second  ;  required  the  direc- 
tion and  velocity  of  their  centre  of 
gravity. 

Ans.  Velocity  is  29.814  feet  per  second. 
Direction  is  E.  26°  33'  54"  X. 


CHAPTER    V. 


ELASTICITY. 

92.  Elastic  and  Inelastic  Bodies.— Elastic  bodies  are 
those  which,  when  compressed,  or  in  any  way  altered  in  form,  tend 
to  return  to  their  original  state.  Those  which  show  no  such 
tendency  are  called  inelastic  or  non-elastic.  No  substance  is 
known  which  is  entirely  destitute  of  the  property  of  elasticity  ; 
but  some  have  it  in  so  small  a  degree  that  they  are  called  in- 
elastic, such  as  lead  and  clay.  Elasticity  is  perfect  when  the 
restoring  force,  whether  great  or  small,  is  equal  to  the  compress- 
ing force.  Air,  and  the  gases  generally,  seem  to  be  almost  per- 
fectly elastic  ;  ivory,  glass,  and  tempered  steel,  are  imperfectly, 
though  highly,  elastic  ;  and  in  different  substances,  the  property 
exists  in  all  conceivable  degrees  between  the  above-named  limits. 


FIG. 


93.  Collision. — Mode   of    Experimenting. — Experiments 
on  collision  may  be  made  with 

balls  of  the  same  density  sus- 
pended by  long  threads,  so  as 
to  move  in  the  line  which 
joins  their  centres  of  gravity. 
If  the  arcs  through  which  they 
swing  are  short  compared 
with  their  radii,  the  balls,  let 
fall  from  different  heights,  will 
reach  the  bottom  sensibly  at 
the  same  time,  and  will  im- 
pinge ivith  velocities  which  are 
very  nearly  proportional  to  the 
arcs.  Thus  A  (Fig.  57),  fall- 
ing from  6,  and  B  from  3,  will  come  into  collision  at  0,  with  veloci- 
ties which  are  as  2  :  1. 

94.  Collision  of  Inelastic  Bodies. — Such  bodies,  after  im- 
pact, move  together  as  one  mass. 

The  velocity  of  tivo  inelastic  bodies  after  collision  is  equal  to  the 
algebraic  sum  of  their  momenta,  divided  by  the  sum  of  their  masses. 

Let  A,  B,  represent  the  masses  of  the  two  bodies,  and  a,  b, 
their  respective  velocities.  Considering  a  as  positive,  if  B  moves 


60  MECHANICS. 

in  the  opposite  direction,  its  velocity  must  be  called  —  b.  Let  v 
be  the  common  velocity  after  impact,  and  suppose  the  bodies  to 
"be  moving  in  the  same  direction,  the  momentum  of  A  is  A  a  ; 
that  of  B  is  B  b  ;  and  the  momentum  of  both  after  collision  is 
(A  +  B)  v~  According  to  the  third  law  of  motion  (Art.  13), 
whatever  momentum  A  loses,  B  gains,  so  that  the  whole  momen- 
tum is  the  same  after  collision  as  before  ;  therefore 


To  find  the  gain  or  loss  of  velocity  of  either  body  subtract  the 
resulting  velocity  from  the  original  velocity  ;  a  negative  result 
indicates  motion  in  a  direction  opposite  to  the  original  motion. 

95.  Questions  on  Inelastic  Bodies.  — 

1.  A,  weighing  3  oz.,  and  moving  10  feet  per  second,  overtakes 
B,  weighing  2  oz.,  and  moving  3  feet  per  second  ;  what  is  the  com- 
mon velocity  after  impact  ?  Ans.  7£  feet  per  second. 

2.  A  weight  of  7  oz.,  moving  11  feet  per  second,  strikes  upon 
another  at  rest   weighing  15   oz.  ;  required  the  velocity  after 
impact  ?  Ans.  3£  feet  per  second. 

3.  A  weighs  4  and  B  2  pounds  ;  they  meet  in  opposite  direc- 
tions, A  with  a  velocity  of  9,  and  B  with  one  of  5  feet  per 
second  ;  what  is  the  common  velocity  after  impact  ? 

-Ans.  4£  feet  per  second. 

4.^4  =  7  pounds,  B  =  4  pounds  ;  they  move  in  the  same 
direction,  with  velocities  of  9  and  2  feet  per  second  ;  required  the 
velocity  lost  by  A  and  gained  by  5?  Ans.  A  2^,  B  4-jSj-. 

5.  A  body  moving  7  feet  per  second,  meets  another  moving  3 
feet  per  second,  and  thus  loses  half  its  momentum  ;  what  are 
the  relative  masses  of  the  two  bodies  ? 

Ans.    A  :  B  ::  13  :  7. 

6.  A  weighs  6  pounds  and  £5  ;  B  is  moving  7  feet  per  second 
in  the  same  direction  as  A  ;  by  collision  B's  velocity  is  doubled  ; 
what  was  A:s  velocity  before  impact  ? 

Ans.  19f  feet  per  second. 

7.  A  body  weighing  100  Ibs.,  and  having  velocity  40  feet  per 
second  meets  another  weighing  20  Ibs.,  and  having  velocity  of  200 
feet  per  second  ;  what  will  be  the  velocity  after  impact  ? 

Ans.  0. 

96.  Collision  of  Elastic  Bodies.  —  Elastic  bodies  after  col- 
lision do  not  move  together,  but  each  has  its  own  velocity.    These 
velocities  are  found  by  doubling  the  loss  and  gain  of  inelastic 
bodies.     When  the  elastic  body  A  impinges  on  B,  it  loses  velocity 


QUESTIONS    ON    ELASTIC    BODIES.  61 

while  it  is  becoming  compressed,  and  again,  while  recovering  its 
form,  it  loses  as  much  more,  because  the  restoring  force  is  equal  to 
the  compressing  force.  For  a  like  reason,  B  gains  as  much  velo- 
city while  recovering  its  form  as  it  gained  while  being  compressed 
by  the  action  of  A.  Hence,  doubling  the  expressions  for  loss  and 
gain  found  by  Art.  94,  and  applying  them  to  the  original  veloci- 
ties, we  find  the  velocity  of  each  body  after  collision,  on  the  sup- 
position of  perfect  elasticity. 

Two  equal  elastic  bodies,  A  and  B,  weighing  50  Ibs.  each, 
moving  with  velocities,  A  =.  40  ft.,  and  B  =  20  ft.  per  second, 
meet ;  what  will  be  the  velocity  of  each  after  impact?  First  we 
must  find  the  gain  and  loss  of  velocity  on  the  supposition  that 
the  bodies  are  inelastic,  and  then  double  such  gain  or  loss  ;  there- 
fore, according  to  Art.  94,  calling  the  velocity  after  impact  v,  we 
have  (50  +  50)  v  =  50  x  40  —  50  x  20,  calling  the  velocity  of 
,B  negative  as  the  bodies  move  in  opposite  directions, — 

1000 

whence  v  =          —  10. 
100 

A  loses  40  —  10  =  30  ft.  per  second,  and  B  loses  —  20  — 
10  =  —  30  feet  per  second  ;  that  is  to  say,  B  loses  all  its  motion 
in  its  original  direction,  and  moves  backward  with  velocity  10. 

Now  as  these  are  elastic  we  must  double  the  gain  and  loss,  and 
we  have  A's  loss  =  60  ft.  and  B's  =  —  60  ft. ;  therefore  A  must 
move  with  velocity  40  —  60  =  —  20,  and  B  with  velocity  —  20 
—  ( —  60)  =  40,  hence  A  must  now  move  in  a  direction  opposite 
to  the  first,  with  velocity  20,  and  B  also  in  direction  opposite  to 
its  previous  motion,  with  velocity  40.  Each  body  takes  the  velocity 
of  the  other  when  the  bodie 


97.  Questions  on  Elastic  Bodies. — 

1.  A,  weighing  10  Ibs.  and  moving  8  feet  per  second,  impinges 
on  B,  weighing  6  Ibs.  and  moving  in  the  same  direction,  5  feet 
per  second  ;  what  are  the  velocities  of  A  and  B  after  impact  ? 

Ans.  A's  =  5f,  B's  =  8f 

2.  A  :  B  : :  4  :  3  ;  directions  the  same ;  velocities  5:4;  what 
is  the  ratio  of  their  velocities  after  impact  ?  Ans.  29  :  36. 

3.  A,  weighing  4  Ibs.,  velocity  6,  meets  B,  weighing  8  Ibs., 
velocity  4  ;  required  their  respective  directions  and  velocities  after 
collision  ?  Ans.  A  is  reflected  back  with  a  velocity  of  ?£, 

and  B  with  a  velocity  of  2f . 

4.  A  and  B  move  in  opposite  directions ;  A  equals  4  B,  and 
I  =  2  a  ;  how  do  the  bodies  move  after  collision  ? 

Ans.  A  returns  with  J,  B  with  If  its  original  velocity. 


62  MECHANICS. 

98.  Impact  on  an  Immovable  Plane.— If  an  inelastic  body 
strikes  a  plane  perpendicularly,   its  velocity  is  simply  destroyed  ; 
its  energy,  however,  is  transformed  into  heat.    If  it  strikes  obliquely,, 
and  the  plane  is  smooth,  it  slides  along  the  plane  with  a  dimin- 
ished velocity.     Let  A  L  (Fig.  58)  Fje 

represent  the  velocity  of  the  body 
before  impact  on  the  plane  P  N, 
and  resolve  it  into  A  C,  perpen- 
dicular, and  C  L,  parallel  to  the 
plane.  Then  A  C,  as  before,  is 
destroyed,  but  C  L  is  not  affected  ; 
hence  its  velocity  on  the  plane 
equals  its  former  velocity  times  the  cosine  of  the  inclination. 

If  a  perfectly  elastic  body  impinges  perpendicularly  upon  a- 
plane,  then,  after  its  velocity  is  destroyed,  the  force  by  which  ifc 
resumes  its  form  causes  an  equal  velocity  in  the  opposite  direction  ; 
that  is,  the  body  rebounds  in  its  own  path  as  swiftly  as  it  struck. 
But  if  the  impact  is  oblique,  the  body  rebounds  at  an  equal  angle 
on  the  opposite  side  of  the  perpendicular.  For,  resolve  A  L, 
as  before,  into  A  C,  C  L  ;  the  latter  continues  uniformly  ;  but, 
instead  of  the  component  A  C,  there  is  an  equal  motion  in  the 
opposite  direction.  Therefoi-e,  if  L  D  is  made  equal  to  C  L,  and 
D  E  equal  to  A  (7,  the  resultant  of  L  D  and  D  E  is  L  E,  which 
is  equal  to  A  L,  and  has  the  same  inclination  to  the  plane. 
Hence,  the  angles  of  incidence  and  reflection  are  equal,  and  on 
opposite  sides  of  the  perpendicular  to  the  surface  at  the  place  of 
impact. 

99.  Imperfect    Elasticity. — The    formulae  for  the    velocity 
of  bodies  after  collision,  and  the  statements  of  the  preceding  arti- 
cle, are  correct  only  on  the  supposition  that  bodies  are,  on  the 
one  hand,  entirely  destitute  of  elasticity,  or  on  the  other  perfectly 
elastic.     As  no  solid  bodies  are  known,  which  are  strictly  of  either 
class,  these  deductions  are  found  to  be  only  near  approximations 
to  the  results  of  experiment.     In  all  practical  cases  of  the  impact 
of  movable  bodies,  the  loss  and  gain  of  velocity  are  greater  than  if 
they  were  inelastic,  and  less  than  if  perfectly  elastic.     And  in  cases 
of  impact  on  a  plane,  there  is  always  some  velocity  of  rebound, 
but  less  than  the  previous  velocity  ;  and  therefore,  if  the  collision 
is  oblique,  the  body  has  less  velocity,  and  makes  a  smaller  angle 
with  the  plane  than  before.     For,  making  D  F  less  than  A  C,  the 
resultant  L  F  is  less  than  A  L,  and  the  angle  D  L  F  is  smaller  than 
D  L  E,  or  A  L  C. 

The  kinetic  energy  of  colliding  elastic  bodies  is  preserved  after 
impact.  Inelastic  bodies  lose  a  portion,  which  appears  as  heat. 


ELASTICITY. 


63 


FIG  59. 


100.  Elasticity  of  Traction. — Bodies,  in  the  form  of  bars 
or  wires,  when  fastened  vertically  at  their  upper 
ends,  and  with  weights  applied  at  their  lower  ends, 
suffer  longitudinal  expansion  (Fig.  59).  Upon  re- 
moving the  weights  the  bodies  resume  their  former 
dimensions  in  virtue  of  their  elasticity,  providing 
the  weights  are  not  too  great.  If,  after  adding 
successively  heavier  weights,  a  body  fails  to  return 
to  its  former  dimensions,  it  is  said  to  have  been 
stretched  beyond  the  limits  of  elasticity. 

Experiments  show  that  (within  the  limits  of  elas- 
ticity) for  elasticity  of  traction 

The  alteration  in  length  is  in  proportion  to  the 
total  length  and  the  acting  load,  and  is  inversely  as  the 
cross-section  of  the  body. 

It  is  also  dependent  upon  a  constant  pertaining 
to  the  substance  of  the  body,  termed  the  modulus 
of  elasticity.  The  relations  are  best  seen  by  employ- 
ing a  formula.  Let 

L  =  length  of  body  ; 

I    =  increase  of  length  by 

load  ; 

P  =  load  in  kg.  ; 
Q  =  cross  -  section     in     sq. 

mm.  ; 
\L  =  modulus. 

I  P  L 


FIG.  60. 


Then 


,  whence^ 


PL 


The  modulus  of  elasticity  is  the  num- 
ber of  kilograms  of  load  required  to  dou- 
ble the  length  of  a  body,  if  its  cross-section 
be  one  square  millimeter. 

Some  substances,  when  strained  or 
compressed,  exhibit  a  property  termed 
elastic  fatigue.  The  alteration  of  length 
increases  somewhat,  if  the  load  be  al- 
lowed to  act  for  some  time.  This  then 
would  give  two  moduli.  In  technical 
tables  the  modulus  for  instantaneous 
elongation  is  given.  After  continued 
load,  the  same  bodies  exhibit  fatigue  in 
returning  to  their  former  shape. 

101.  Elasticity  of  Torsion.— If 
a  weight,  suspended  by  a  wire  (Fig.  60) 


64  MECHANICS. 

and  supplied  with  an  index  moving  over  a  graduated  circle,  be 
twisted  through  an  angle  and  then  released,  the  torsional  elasticity 
of  the  wire  will  cause  it  to  turn  back  into  its  former  position.  The 
inertia  of  the  weight  will  cause  a  twist  in  an  opposite  direction 
and  the  index  will  continue  to  describe  a  series  of  very  nearly 
isochronous  oscillations  about  the  point  of  equilibrium. 

Experiments  upon  elasticity  of  torsion  have  shown  that  the 
amount  of  force  exerted  by  a  twisted  wire  is  directly  proportional  to 
the  angle  through  which  it  has  been  twisted. 

This  fact  is  of  great  value,  as  it  lies  at  the  foundation  of  the 
construction  of  many  instruments  for  exact  physical  measurement. 

102.  Elasticity  of  Flexure. — A  substance,  in  the  form  of 
a  bar,  clamped  at  one  end,  so  as  to  lie  in  a  horizontal  position, 
will  be  deflected  (Fig.  61)  from  the  horizontal  if  a  weight  be  ap* 

FIG.  61. 


plied  at  the  free  end.    The  amount  of  the  deflection  of  the  free  end,, 

_4PT 
-fTa^' 

where  I  =  length  of  free  bar,  P  =  applied  weight,  //,  =  modulus  of 
elasticity,  a  =  vertical  side,  and  b  =  horizontal  breadth. 

Elasticity  of  flexure  finds  many  practical  applications.  Watch- 
springs,  carriage  springs,  spring  balances,  dynamometers,  all  spiral 
springs  are  dependent  upon  elasticity  of  flexure  for  their  action. 


CHAPTEE    VI. 

SIMPLE     MACHINES. 

103.  Classification  of  Machines. — In  the  preceding  chap- 
ters the  motion  of  bodies  has  been  supposed  to  arise  from  the 
immediate  action  of  one  or  more  forces.  But  a  force  may  pro- 
duce effects  indirectly,  by  means  of  something  which  is  inter- 
posed for  the  purpose  of  changing  the  mode  of  action.  These 
intervening  bodies  are  called,  in  general,  machines;  though  the 
names,  tools,  instruments,  engines,  &c.,  are  used  to  designate  par- 
ticular classes  of  them.  The  elements  of  machinery  are  called 
simple  machines.  The  following  list  embraces  those  in  most 
common  use  : 

1.  The  lever. 

2.  The  wheel  and  axle. 

3.  The  pulley. 

4.  The  rope  machine. 

5.  The  inclined  plane. 
G.  The  wedge. 

7.  The  screw. 

8.  The  knee-joint. 

In  respect  to  principle,  these  eight,  and  all  others,  may  be 
reduced  to  three. 

1.  The   law  of  equal  moments,   applicable   in   those   cases   in 
which  the  machine  turns  on  a  pivot  or  axis,  as  in  the  lever  and 
the  wheel  and  axle. 

2.  The  principle  of  transmitted  tension,  to  be  applied  wherever 
the  force  is  exerted  through  a  flexible  cord,  as  in  the  pulley  or 
rope  machine. 

3.  The  principle  of  oblique  action,  applicable  to  all  the  other 
machines,  the  force  being  employed  to  balance  or  overcome  one 
component  ouly  of  the  resistance. 

The  force  which  transmits  its  energy  to  the  machine  is  called 
the  power.  This  word  is  not  to  be  confused  with  power  as  denned 
in  Art.  38.  From  long  usage  by  various  text-book  authors,  the 
word  has  become  so  firmly  attached  to  machines  that  it  does  not 
seem  advisable  to  discard  it. 

The  force  which  resists  the  power,  and  is  balanced  or  overcome 
by-it,  is  called  the  weight. 

A  compound  machine  is  one  in  which  two  or  more  simple  ma- 
chines are  so  connected  that  the  weight  of  the  first  constitutes  the 
power  of  the  second,  the  weight  of  the  second  the  power  of  the 
third,  &c. 

5 


66  MECHANICS. 

I.  THE  LEVEB. 

104.  The  Three  Orders  of  Straight  Lever.— The  lever 
is  a  bar  of  any  form,  free  to  turn  on  a 
fixed  point,  which  is  called  the  ful-     ^  c        £ 

crum.     In  the  first  order  of  lever,  the         r s 1 

fulcrum  is  between  the  power  and     -^  F 

weight  (Fig.  62)  ;  in  the  second,  the  <P 

weight  is  between  the  power  and  ful- 
crum (Fig.  63)  ;  in  the  third,  the  power  is  between  the  weight 
and  fulcrum  (Fig.  64). 

FIG.  63.  FIG.  04- 


105.  Equal  Moments  in  Relation  to  the  Fulcrum.— 

According  to  the  principle  of  moments,  we  find  for  each  order 
of  the  lever,  P  x  A  0  —  W  x  B  C  ;  that  is, 

The  power  and  weight  have  equal  moments  in  relation  to  the 
fulcrum. 

The  moment  of  either  force  is  the  measure  of  its  efficiency  to 
turn  the  lever  ;  for,  since  the  lever  is  in  equilibrium,  the  efficiency 
of  the  power  to  turn  it  in  one  direction  must  equal  the  efficiency 
of  the  weight  to  turn  it  in  the  opposite  direction.  We  may 
therefore  use  P  x  A  C  to  represent  the  former,  and  W  x  B  (7, 
the  latter. 

If  several  forces,  as  in  Fig.  65,  are  in  equilibrium,  some  tending 

FIG,  65. 
ABC  D 


to  turn  the  bar  in  one  direction,  and  others  in  the  opposite,  then 
A  and  B  must  have  the  same  efficiency  to  produce  one  motion  as 
C  and  D  have  to  produce  the  opposite ;  that  is,  Ax  A  G  +  Bx.B  G 
=  CxCG+DxDG;  or, 

The  sum  of  the  moments  of  A  and  B  equals  the  sum  of  the  mo- 
ments of  C  and  D. 


Dk 


ACTING    DISTANCE.  67 

In  order  to  allow  for  the  influence  of  the  weight  of  the  lever 
itself,  consider  it  to  be  collected  at  its  centre  of  gravity,  and  add 
its  moment  to  that  of  the  power  or  weight,  according  as  it  aids 
the  one  or  the  other.  In  Fig.  62,  let  the  weight  of  the  lever  =  w, 
and  the  distance  of  its  centre  from  (7  on  the  side  of  P  =  in  ;  then 
P  x  A  C  +  m  w  =  W  x  B  C.  In  the  2d  and  3d  orders,  the  mo- 
ment of  the  lever  necessarily  aids  the  weight  ;  and  hence,  in  each 
case,  P  x  AC  =  W  x  BC  +  mio. 

If  a  weight  hangs  on  a  bar  between  two  supports,  as  in  Fig.  66, 
it  may  be  regarded  as  a  lever  of  the 
2d  order,  the  reaction  of  either  sup- 

port being  considered  as  a  power.     J^  ?          f 

Let  F  denote  the  reaction  at  A,  and     f 
F'  at  C;   then  by  the  theorems  of 
parallel  forces,  we  have  the  pressures 
at  A  and  C  inversely  as  their  dis- 
tances  from  B,  and  W  =  F  +  F'. 

Resuming  the  equation  P  x  A  C  =  W  x  B  C,  we  derive  the 
proportion  P  :  W  ::  B  C  :  A  C  ;  hence,  in  each  order  of  the 
straight  lever,  when  the  forces  act  in  parallel  lines,  The  power 
and  weight  are  inversely  as  the  lengths  of  the  arms  on  which 
they  act. 

106.  The  Acting  Distance.  —  In  the  three  orders,  as  above 
described,  the  equilibrium  is  not  destroyed  by  inclining  the  lever 
to  any  angle  whatever  with  the  horizon,  provided  the  lever  is  sym- 
metrical with  respect  to  its  longer  axis  and  the  centre  of  motion 
C  is  on  this  axis  and  not  above  or  below  it,  and  provided  the  direc- 
tions of  the  forces  remain  vertical.  For  by  the  principle  of 
parallel  forces  any  straight  line  intersecting  the  lines  of  the  forces 
is  divided  by  the  line  of  the  resultant 
into  parts  which  are  inversely  as  the  IG< 


forces  ;  therefore  (Fig.  67)  b  C  :  a  C  :  :      A  M 
P  :  W.     Hence,  the  resultant  of  P 
and  W  remains  at  C,  in  every  position 
of  the  lever.     By  similar  triangles, 
bC:  aC  ::   ON:  CM-,  .;.  P:  W  :: 
ON:  CM;   .-.  PxCM=WxCN.      • 
The  lines   CM  and  ON,  which  are         * 


drawn  from  the  fulcrum  perpendic- 

ular to  the  lines  in  which  the  forces  act,  are  called  the  acting  dis- 
tances or  the  lever  arms  of  the  power  and  weight,  respectively. 
And  as  they  may  be  employed  in  levers  of  irregular  form,  the 
moments  of  power  and  weight  are  usually  measured  by  the  pro- 
ducts, P  x  C'J/and  W  x  C  N\  therefore,  the  power  multiplied 


68  MECHANICS. 

by  its  acting  distance  equals  the  weight  multiplied  ~by  its  acting 
distance ;  or,  more  briefly,  the  moment  of  the  power  equals  the 
moment  of  the  weight,  as  in  Art.  105.  In  Figs.  62,  63,  and  64, 
the  acting  distances  are  in  each  case  identical  with  the  arms  of 
the  lever. 

107.  Lever  not  Straight,  and  Forces  not  Parallel. — 
Let  A  C  B  (Fig.  68)  be  a  lever  of  any  form,  and  let  it  be  in  equi- 
librium by  the  forces  P  and  P', 

acting  in  any  oblique  directions 
in  the  same  plane.  Produce  P  A 
and  P'  B  till  they  meet  in  D ; 
then,  if  the  fulcrum  is  at'C",  the 
resultant  must  be  in  the  direction 
D  C ';  otherwise  the  reaction  of 
the  fulcrum  cannot  keep  the  sys- 
tem in  equilibrium  (Art.  43). 
Therefore  (Art  44), 

P  :  P'  : :  sin  B  D  G  :  sin  A  D  C. 

Draw  C  M  perpendicular  to  A  D,  and  C  N  to  B  D,  and  they 
are  the  sines  of  A  D  C  and  B  D  C}  to  the  same  radius  D  C. 

.-.  P  :  P'  ::  ON:  CM;  and  P  x  CM=  P1  x  C  N. 

The  lines  C  M  and  C  N  are  the  acting  distances  of  P  and  P'  • 
therefore  the  law  of  the  lever  in  all  cases  is  the  same,  namely  : 

The  moment  of  the  power  equals  the  moment  of  the  weight. 

When  the  forces  act  obliquely,  the  pressure  on  the  fulcrum  is 
less  than  the  sum  of  the  forces ;  for,  if  C  E  is  parallel  to  B  D, 
then  D  E,  E  C,  and  C  D,  represent  the  three  forces  which  are  in 
equilibrium.  But  C  D  is  less  than  the  sum  of  D  E  and  E  C. 

108.  The  Compound   Lever. — When  a  lever  acts  on  a 
second,  that  on  a  third,  &c.,  the  machine  is  called  a  compound 
lever.    The  law  of  equilibrium  is — 

The  continued  product  of  the  power  and  acting  distances  on  the 
side  of  the  power  is  equal  to  the  continued  product  of  the  weight 
and  acting  distances  on  the  side  of  the  weight. 

Let  the  force  exerted  by  A  B  on  B  D  (Fig.  69)  be  called  x,  and 
that  of  B  D  on  D  E  be  called  y  ;  then 

PxAC=xxBC; 
x  x  B F  —  y  x  DP; 
y  x  DG  =W  x  GE. 


THE    BALANCE.  69 

Multiply  these  equations  together  and  omit  common  factors,  and 
we  have 

PxACxBFxDQ=WxBCxDFxGE. 

FIG.  69. 


G     JE 


I 


If  the  levers  were  of  irregular  forms,  the  acting  distances  might 
not  be  identical  with  the  arms,  as  they  are  in  the  figure. 

109.  The  Balance.— This  is  a  common  and  valuable  instru- 
ment for  weighing.  It  is  a  straight  lever  with  equal  arms,  having 
scale-pans,  either  suspended  at  the  ends,  or  standing  upon  them, 
one  to  contain  the  poises,  and  the  other  the  substance  to  be 
weighed.  For  scientific  purposes,  particularly  for  chemical  analy- 
sis, great  care  is  bestowed  on  the  construction  of  the  balance. 

The  arms  of  the  balance,  measured  from  the  fulcrum  to  the 
points  of  suspension,  must  be  precisely  equal. 

The  knife-edges  forming  the  fulcrum,  and  the  points  of  sus- 
pension, are  made  of  hardened  steel,  and  arranged  exactly  in  a 
straight  line. 

The  centre  of  gravity  of  the  beam  is  beloiv  the  fulcrum,  so  that 
there  may  be  a  stable  equilibrium  ;  and  yet  below  it  by  an  exceed- 
ingly sihall  distance,  in  order  that  the  balance  may  be  very 
sensitive. 

To  preserve  the  edge  of  the  fulcrum  from  injury,  the  beam  is 
raised  by  supports  called  Y's,  when  not  in  use. 

A  long  index  at  right  angles  to  the  beam,  points  to  zero  on  a 
scale  when  the  beam  is  horizontal. 

To  protect  the  instrument  from  dust  and  moisture  at  all  times, 
and  from  air-currents  while  weighing,  the  balance  is  in  a  glass 
case,  whose  front  can  be  raised  or  lowered  at  pleasure. 

A  balance  for  chemical  analysis  is  shown  in  Fig.  70.  By  turn- 
ing the  knob  0.  the  beam  can  be  raised  on  the  Y's  A  A  from  the 
surface  on  which  the  fulcrum  K  rests.  The  screw  C  raises  and 
lowers  the  fulcrum  in  relation  to  the  centre  of  gravity  of  the  beam, 
in  order  to  increase  or  diminish  the  sensitiveness  of  the  instru- 
ment. In  the  most  carefully  made  balances,  the  index  will  make 


70 


MECHANICS. 


a  perceptible  change,  by  adding  to  the  scale  one  millionth  of 
poise. 

FIG.  70. 


For  commercial  purposes,  it  is  convenient  to  have  the  scale- 
pans  above  the  beam.  This  is  done  by  the  use  of  additional  bars, 
which  with  the  beam  form  parallelograms,  whose  iipright  sides 
are  rods,  projecting  upward  and  supporting  the  scales.  Such  con- 
trivances necessarily  increase  friction  ;  but  balances  so  constructed 
are  sufficiently  sensitive  for  ordinary  weighing. 

110.  The  Steelyard. — This  is  a  weighing  instrument,  hav- 
ing a  graduated  arm,  along  which  a  poise  may  be  moved,  in  order 
to  balance  various  weights  on  the  short  arm.  While  the  moment 
of  the  article  weighed  is  changed  by  increasing  or  diminishing  its 
quantity,  that  of  the  poise  is  changed  by  altering  its  acting  dis- 
tance. Since  P  x  A  C  =  W  X  B  C  (Fig.  71),  and  P  is  constant, 

FIG.  71. 


C  E 


and  also  the  distance  B  C  constant,  A  C  <x  W;  hence,  if  W  is 
successively  1  lb.,  2  Ibs.,  3  Ibs.,  &c.,  the  distances  of  the  notches, 


PLATFORM     SCALES.  71 

a,  b,  c,  &c.,  are  as  1,  2,  3,  &c.  ;  in  other  words,  the  bar  CD  is 
divided  into  equal  parts.  In  this  case,  the  graduation  begins 
from  the  fulcrum  C  as  the  zero  point. 

But  suppose,  what  is  often  true,  that  the  centre  of  gravity  of 
the  steelyard  is  on  the  long  arm,  and  that  P  placed'  at  E  would 
balance  it ;  then  the  moment  of  the  instrument  itself  is  on  the 
side  C  D,  and  equals  P  x  C  E.     Hence,  the  equation  becomes 
P  x  A  C  +  P  x  C  E  =  W  x  B  C;  or 
P  x  A  E  =  W  x  B  C. 

„•.  W  oc  A  E ;  and  the  graduation  must  be  considered  as  com- 
mencing at  E  for  the  zero  point.  Such  a  steelyard  cannot  weigh 
below  a  certain  limit,  corresponding  to  the  first  notch  a. 

To  find  the  length  of  the  divisions  on  the  bar,  divide  A  E,  the 
distance  of  the  poise  from  the  zero  point,  by  W,  the  number  of 
units  balanced  by  P,  when  at  that  distance. 

The  steelyard  often  has  two  fulcrums,  one  for  less  and  the  other 
for  greater  weights. 

111.  Platform  Scales. — This  name  is  given  to  machines 
arranged  for  weighing  heavy  and  bulky  articles  of  merchandise. 
The  largest,  for  cattle,  loaded  wagons,  &c.,  are  constructed  with 
the  platform  at  the  surface  of  the  ground.  In  order  that  the  plat- 
form may  stand  firmly  beneath  its  load,  it  rests  by  four  feet  on  as 
many  levers  of  the  second  order,  whose  arms  have  equal  ratios. 
A  F,  B  F,  C  G,  D  G  (Fig.  72),  ar.e  four  such  levers,  resting  on  the 

FIG.  72. 


fulcrums,  A,  B,  C,  D,  while  the  other  ends  meet  on  the  knife- 
edge,  F  G,  of  another  lever,  L  M.  This  fifth  lever  has  its  fulcrum 
at  L,  and  its  outer  extremity  is  attached  by  a  vertical  rod,  M  N, 
to  a  steelyard,  whose  fulcrum  is  E,  and  poise  P.  The  five  levers 
are  arranged  in  a  square  cavity  just  below  the  surface  of  the 
ground.  The  dotted  line  shows  the  outline  of  the  cavity.  On  the 
bearing  points  of  the  four  levers,  H,  1,  J,  K,  rest  the  feet  of  the 
platform  (not  represented),  which  is  firmly  built  of  plank,  and  just 


—   1 


72  MECHANICS. 

fits  into  the  top  of  the  cavity  without  touching  the  sides.  The 
machine  is  a  compound  lever  of  three  parts  ;  for  the  four  levers 
act  as  one  at  F  G,  and  are  used  to  give  steadiness  to  the  platform 
which  rests  upon  them. 

A  construction  quite  similar  to  the  above  is  made  of  portable 
size,  and  used  in  all  mercantile  establishments  for  weighing  heavy 
goods. 

112.  Questions  on  the  Lever. — 1.  A  B  (Fig.  73)  is  a 
uniform  bar,  2  feet  long,  and 

weighs  4  oz. ;  where  must  the  f  ul-  B  • IG<  73'      c 

crum  be  put,  that  the  bar  may  be  _— 
balanced  by  P,  weighing  5  Ibs.  ? 
Am.  ^  of  an  inch  from  A. 

2.  A  lever  of  the  second  order 

is  25  feet  long  ;  at  what  distance  from  the  fulcrum  must  a  weight 
of  125  pounds  be  placed,  so  that  it  may  be  supported  by  a  power 
able  to  sustain  60  pounds,  acting  at  the  extremity  of  the  lever. 

Ans.  12  feet. 

3.  A  and  B  are  of  the  same  height,  and  sustain  upon  their 
shoulders  a  weight  of  150  pounds,  placed  on  a  pole  9£  feet  long  ; 
the  weight  is  placed  6|  feet  from  A  ;  what  is  the  weight  sustained 
by  each  person  ? 

Ans.  A  sustains  42$  Ibs.,  and  B  sustains  107-f  Ibs. 

4.  A  bent  lever,  A  C  B  (Fig.  74),  has  the  arm  A  C  =  3  feet,. 
C  B  =  8  feet,   P  =  5  Ibs.,  and  the 

angle  A  C  B  =  140°  ;  what  weight, 
W,  must  be  attached  at  B,  in  order  to 
keep  A  C  horizontal  ? 

Ans.  2.4476  Ibs. 

5.  A  cylindrical  straight  lever  is 
14  feet  long,  and  weighs  6  Ibs.  5  oz. ; 
its  longer  arm  is  9,  and  its  shorter  5 
feet ;  at  the  extremity  of  its  shorter 
arm  a  weight  of  15  Ibs.  2  oz.  is  sus- 
pended ;  what  weight  must  be  placed 

at  the  extremity  of  the  longer  arm  to  W 

keep  it  in  equilibrium?  Ans.  7  Ibs. 

6.  A  uniform  bar,  12  feet  long,  weighs  7  Ibs. ;  a  weight  of  10> 
Ibs.  hangs  on  one  end,  and  2  feet  from  it  is  applied  an  upward 
force  of  25  Ibs.,  where  must  the  fulcrum  be  put  to  produce  equi- 
librium ?  Ans.  1  foot  from  the  10  Ibs. 

7.  The  lengths  of  the  arms  of  a  balance  are  a  and  b.     When  p 
ounces  are  hung  on  a,  they  balance  a  certain  body ;  but  it  re- 


WHEEL    AND    AXLE. 


73 


quires  q  ounces  to  balance  the  same  body,  when  placed  in  the 
other  scale.  What  is  the  true  weight  of  the  body  ?  According  to 
the  first  weighing,  a  p  •=  b  x ;  according  to  the  second,  b  q  =  a  x. 
/.  a  b  p  q  :  =  a  b  x\  and  x  =  V  p  q-  Hence,  the  true  weight  is 
a  geometrical  mean  between  the  apparent  weights. 

8.  On  one  arm  of  a  false  balance  a  body  weighs  11  Ibs.,  on  the 
other,  17  Ibs.  3  oz.;  what  is  the  true  weight  ? 

Ans.  13  Ibs.  12  oz. 

9.  Four  weights  of  1,  3,  5,  7  Ibs.,  respectively,  are  suspended 
from  points  of  a  straight  lever,  eight  inches  apart ;  how  far  from 
the  point  of  suspension  of  the  first  weight  must  the  fulcrum  be 
placed,  that  the  weights  may  be  in  equilibrium  ? 

Ans.  17  inches. 

10.  Two  weights  keep  a  horizontal  lever  at  rest,  the  pressure 
on  the  fulcrum  being  10  Ibs.,  the  difference  of  the  weights  4  Ibs., 
and  the   difference   of  the  lever  arms  9  inches  ;   what  are  the 
weights  and  their  lever  arms  ? 

Ans.  Weights,  7  Ibs.  and  3  Ibs.;  arms,  6|  in.  and  15f  in. 

II.  THE  WHEEL  AND  AXLE. 

113.  Description  and  Law  of  the  Machine. — The  wheel 
and  axle  consists  of  a  cylinder  and  a  wheel,  firmly  united,  and  free 
to  revolve  on  a  common  axis.  The  power  acts  at  the  circumfer- 
ence of  the  wheel  in  the  direction  of  a  tangent,  and  the  weight  in 
the  same  manner,  at  the  circumference  of  the  cylinder  or  axle ;  so 
that  the  acting  distances  are  the  radii  at  the  two  points  of  contact. 
As  the  system  revolves,  the  radii  successively  take  the  place  of 
acting  distances,  without  altering  at  all  the  relation  of  the  forces 
to  each  other.  The  wheel  and  axle  is  therefore  a  kind  of  endless 
lever. 

Let  W  (Fig.  75)  be  the  weight  suspended  from  the  axle,  tend- 
ing to  revolve  it  on  the  line  L  M ; 
and  P,  the  power  acting  on  *the 
wheel,  tending  to  revolve  the  sys- 
tem in  the  opposite  direction.  It 
is  plain  that  the  acting  distances 
are  the  radius  of  the  axle,  and  A  C 
the  radius  of  the  wheel.  In  case  of 
equilibrium,  the  moment  of  W* 
equals  the  moment  of  P.  Calling 
the  radius  of  the  axle  r,  and  the 
radius  of  the  wheel  R,  then  W  x  r 
—  P  x  R ;  or 

P  :  W  ::r:  R. 


FIG.  75. 


74 


MECHANICS. 


FIG. 


If,  instead  of  the  weight  P,  suspended  on  the  wheel,  the  rope 
be  drawn  by  any  force  in  the  direction  P'  or  P",  it  is  still  tangent 
to  the  circumference,  and  therefore  its  acting  distance,  0  D  or  G  B, 
the  same  as  before.  In  general,  the  law  of  equilibrium  for  this 
machine  is, 

The  moment  of  the  Power  is  equal  to  the  moment  of  the 
Weight. 

If  the  rope  on  the  wheel,  being  fastened  at  A  (Fig.  76)  is 
drawn  by  the  side  of  the  wheel,  as  A  P', 
the  acting  distance  of  the  power  is  dimin- 
ished from  CA  to  C  E,  and  therefore  its 
efficiency  is  diminished  in  the  same  ratio. 
Were  the  rope  drawn  away  from  the 
wheel,  as  A  P",  making  an  equal  angle 
on  the  other  side  of  A  P.  the  same  effect 
is  produced,  the  acting  distance  now  be- 
coming OF. 

Tlie  radius  of  the  ivheel  and  the  radius 
of  the  axle  should  each  be  reckoned  from 
the  axis  of  rotation  to  the  centre  of  the 
rope  ;  that  is,  half  of  the  thickness  of  the  rope  should  be  added  to 
the  radius  of  the  circle  on  which  it  is  coiled.  Calling  t  the  half 
thickness  of  the  rope  on  the  axle,  and  t'  that  of  the  rope  on  the 
wheel,  the  equation  of  equilibrium  is 

P  x  (R  +  t')  =  IF  x  (r  +  t). 

In  considering  the  wheel  and  axle  no  account  has  been  taken 
of  the  stiffness  of  the  rope,  which  acts  as  a  constant  resistance, 
opposing  motion  in  winding  upon  a  drum  or  wheel,  and  also  in 
unwinding. 

114.  Differential  Pulley. — A  modification  of  the  wheel  and 
axle,  called  a  differential  pulley,  is  of  great  use  in  raising  very 
heavy  weights  through  short  distances. 

The  pulley  consists  of  .a  solid  wheel  A  (Fig.  77),  one  half  of 
which,  b,  is  of  less  diameter  than  the  other  half,  a, 
suspended  in  a  block  in  the  usual  manner. 

A  continuous  chain  is  used,  which  we  may  trace 
from  the  point  A  (Fig.  78),  upward,  over  the  larger 
of  the  two  circumferences,  then  downward  through 
B  to  the  movable  pulley  D,  thence  upward  through 
C' around  the  smaller  circumference  of  the  wheel, 
thence  down  through  E  and  back  to  the  point  of 
beginning  at  A. 
Call  the  radii  R  and  r  as  indicated  in  the  figure,  and  suppose 


FIG.  77. 


COMPOUND   WHEEL  AND  AXLE. 


75 


then 


FIG.  78. 


a  downward   force  P  to  be   applied  to  the  chain  at   A, 
PxR  +  $Wxr  =  %WxR,  in  which  equa- 
tion no  account  is  taken  of  the  weight  of  the  chain. 
Transposing,  we  obtain, 

P  x  R  =  |  W  (R—r)  or 

P:|  W  ::  R—r:  R. 

Now  R — r  may  be  made  as  small  as  we  please, 
and  hence  the  power  also  may  be  made  small  as  com- 
pared with  the  weight.  The  weight  of  the  chain 
and  the  friction  act  as  resistances  to  motion,  and  are 
sufficient  to  prevent  the  downward  run  of  D  after 
the  hand  is  removed  from  A,  even  when  W  is  very 
great.  This  pulley  may  be  found  in  any  large 
foundry,  or  machine  shop. 

115.  The  Compound  Wheel  and  Axle. — 

When  a  train  of  wheels,  like  that  in  Fig.  79,  is  put 
in  motion,  those  which  communicate  motion  by  the 
circumference  are  called  driving  wheels,  as  A  and  (7; 
those  which  receive  motion  by  the  circumference 
are  called  driven  wheels.  And  the  law  of  equili- 
brium is, 

The  continued  product  of  the  power  and  radii  of 
the  driven  wheels  is  equal  to  the  continued  product  of  the  weight 
and  radii  of  the  driving  wheels. 

The  crank  P  Q  is  to  be  reckoned  among  driven  wheels ;  the 
axle  E  among  driving  wheels.  FlG 

Let  the  radius  of  B  be  called  R ;  of 
D,  R' ;  of  A,  r;   of  {?,  r' ;  of  E,  r". 
Call  the  force  exerted  by  A  on  B,  «; 
that  of  C  on  D,  y.     Then 
P  x  P  Q=  r  x  x\ 
x  x  R  =  r'  x  y  ; 
y  x  R  =  r"  x   W. 
Multiply  and  omit  common  factors,  and 
we  have 
PxPQxRxR'=  TTx  r"  xr'xr. 

If  the  driving  wheels  are  equal  to  each  other,  and  also  the 
driven  wheels,  and  the  number  of  each  is  n,  then 
P  x  R"  =  W  r\ 

116.  Direction  and  Rate  of   Revolution.— When   two 

wheels  are  geared  together  by  teeth,  they  necessarily  revolve  in 
contrary  directions.  Hence,  in  a  train  of  wheels,  the  alternate 
axles  revolve  the  same  way. 

The  circumferences  of  two  wheels  which  are  in  gear  move  with 


76  MECHANICS. 

the  same  velocity  ;  hence  the  number  of  revolutions  will  be  recip- 
rocally as  the  radii  of  the  wheels. 

Since  teeth  which  gear  together  are  of  the  same  size,  the  rela- 
tive number  of  teeth  is  a  measure  of  the  relative  circumferences, 
and  therefore  of  the  relative  radii  of  the  wheels.  If  the  wheel  A 
(Fig.  79)  has  20  teeth,  and  B  has  40,  and  again  if  C  has  15,  and 
D  45,  then  for  every  revolution  of  B,  A  revolves  twice,  and  for 
every  revolution  of  D,  C  revolves  three  times.  Therefore,  six 
turns  of  the  crank  are  necessary  to  give  one  revolution  to  the 
axle  E. 

By  cutting  the  teeth  of  wheels  on  a  conical  instead  of  a  cylin- 
drical surface,  the  axles  may  be  placed  at  any  angle  with  each 
other,  as  represented  in  Fig.  80. 

Whether  axles  are  parallel  or  not,  bands  in- 
stead of  teeth  may  be  used  for  transmitting 
rotary  motion.  But  as  bands  are  liable  to  slip 
more  or  less,  they  cannot  be  employed  in  cases 
requiring  exact  relations  of  velocity. 

117.  Questions    on  the   Wheel  and 
Axle.— 

1.  A  power  of  12  Ibs.  balances  a  weight  of 
100  Ibs.  by  a  wheel  and  axle ;  the  radius  of  the 
axle  is  6  inches ;  what  is  the  diameter  of  the  wheel  ? 

Ans.  8  ft.  4  in. 

2.  JF=500  Ibs.;  R  =  4  ft;  r  =  8  in.;  the  weight  hangs  by  a 
rope  1  inch  thick,  but  the  power  acts  at  the  circumference  of  the 
wheel  without  a  rope  ;  what  power  will  sustain  the  weight  ? 

Ans.  88.54  Ibs. 

3.  In  Fig.  79,  A  and  C  have  each  15  teeth,  B  and  D  each  40 
teeth  ;  the  radius  of  the  axle  ^is  4  inches;  the  rope  on  it  1  inch 
in  diameter ;  and  the  radius  of  the  crank  P  Q  is  18  inches ;  what 
is  the  ratio  of  power  to  weight  in  equilibrium?  Ans.  1  :  28f. 

118.  The  Pulley  Described. — The  pulley  consists  of  one 
or  more  wheels  or  rollers,  with  a  rope  passing  over  the  edge  in 
which  a  groove  is  sunk  to  keep  the  rope  in  place.     The  axis  of 
the  roller  is  in  a  block,  which  is  sometimes  fixed,  and  sometimes 
rises  and  falls  with  the  weight ;  and  the  pulley  is  accordingly 
called  a  fixed  pulley  or  a  movable  pulley.     The  principle  which 
explains  the  relation  of  power  and  weight  in  every  form  of 
pulley  is  this : 

Whatever  strain  or  tension  is  applied  to  one  end  of  a  cord,  is 
transmitted  through  its  whole  length,  if  it  does  not  branch,  however 
much  its  direction  is  changed. 


THE    FIXED    PULLET. 


FIG.  81. 


FIG.  82. 


In  the  pulley,  the  sustaining  portions  of  the  rope  are  assumed 
to  be  parallel  to  each  other. 

119.  The  Fixed  Pulley.— In 

the  fixe4  pulley,  A  (Fig.  81),  the 
force  P,  produces  a  tension  in  the 
string,  which  is  transmitted  through 
its  whole  length,  and  which  can  be 
balanced  only  when  W  equals  P. 
Hence,  in  the  fixed  pulley,  the 
power  and  weight  are  equal.  This 
machine  is  useful  for  changing  the 
direction  in  which  the  force  is  ap- 
plied to  the  weight:  and  if  the 
power  only  acts  in  the  plane  of  the 
groove  of  the  wheel,  it  is  immaterial  what  is  its  direction,  horizon- 
tal, vertical,  or  oblique. 

120.  The  Movable  Pulley.— In  Fig.  82,  the       FIG.  83. 
tension  produced  by  P,  is  transmitted  from  A  down 

to  the  wheel  E,  and  thence  up  to  D  ;  therefore  W  is 
sustained  by  two  portions  of  the  rope,  each  of  which 
exerts  a  force  equal  to  P. 

.-.  W  =  2  P;   or  P  :  W : :  1  :  2. 

The  same  reasoning  applies,  where  the  rope  passes 
between  the  upper  and  lower  blocks  any  number  of 
times,  as  in  Fig.  83.  The  force  causes  a  tension  in  the 
rope,  which  is  transmitted  to  every  portion  of  it.  If  n 
is  the  number  of  portions  which  sustain  the  lower 
block,  then  W  is  upheld  by  n  P  ;  and  if  there  is  equi- 
librium, P  :  W  : :  1  :  n.  In  the  figure,  the  weight 
equals  six  times  the  power.  The  law  of  equilibrium, 
therefore,  for  the  movable  pulley  with  one  rope,  is  this, 

The  power  is  to  the  weight  as  one  to  the  number  of 
the  sustaining  portions  of  the  rope. 

121.  The  Compound  Pulley.— Wherever  a  system  of  pul- 
leys has  separate  ropes  the  machine  is  to  be  regarded  as  com- 
pound, and  its  efficiency  is  calculated  accordingly.     Figures  84 
and  85  are  examples.     In  Fig.  84  call  the  weight  sustained  by  F,  x> 
and  that  sustained  by  D,  y.     Then  (Art.  120), 

P  :    x::  1  :2; 
x  :    y  : :  1  :  2  : 
y  :  W::  1  :  2. 
/.  P  :  W : :  1  :  23  : :  1  :  8. 
And  if  n  is  the  number  of  ropes,  P  :  W  : :  1  :  2*. 


78 


MECHANICS. 


In  Fig.  85  the  tension  P  is  transmitted  over  A  directly  to  the 
weight  at  G  ;  the  wheel  A  is  loaded,  therefore,  with  2  P,  and  a 
tension  of  2  P  comes 

FlG-  84 


upon  the  second  rope, 
which  is  transmitted 
over  B  to  the  weight  at 
F.  In  like  manner,  a 
tension  of  4  P  is  trans- 
mitted over  C  to  K 
The  sum  of  all  these 
being  applied  to  the 
weight,  it  must  there- 
fore be  equal  to  that 
sum  in  case  of  equili- 
brium. Therefore,  P  : 


Now  the  sum  of  this 

geometrical  series  to  n  terms  is  2"  —  1  , 
.-.  P  :  W ::  1  :  2"  —  1.  This  combination  is 
therefore  a  little  less  efficient  than  the  pre- 
ceding. • 

Since  the  several  ropes  have  different  ten- 
sions, the  weight  cannot  be  balanced  upon 
them,  unless  those  of  greatest  tension  are 
nearest  the  line  of  direction  of  the  body. 
For  example,  if  the  rope  F  is  directed  toward 
the  centre  of  gravity  of  the  weight,  the  rope  G  should  be  attached 
four  times  as  far  from  it  as  the  rope  E,  in  order  to  prevent  the 
weight  from  tipping. 

The  pulley  owes  its  efficiency  as  a  machine  to  the  fact,  that  the 
tension  produced  by  the  power  is  applied  repeatedly  to  the  weight. 
The  only  use  of  the  wheels  is  to  diminish  friction.  Were  it  not 
for  friction,  the  rope  might  pass  round  fixed  pins  in  the  blocks, 
and  the  ratio  of  power  to  weight  would  still  be  in  every  case  the 
same  as  has  been  shown. 


IV.  THE  EOPE  MACHINE. 

122.  Definition  and  Law  of  this  Machine.— 
The  rope  machine  is  one  in  which  the  power  and  weight  c,re  in 
equilibrium  by  the  tension  of  one  or  more  ropes. 

According  to  this  definition  the  pulley  is  included.  It  is  that 
particular  form  of  the  rope  machine  in  which  the  sustaining  parts 
of  the  ropes  are  parallel ;  and  it  is  treated  as  a  separate  machine, 


THE    ROPE    MACHINE. 


79 


FIG.  86. 


because  its  theory  is  very  simple,  and  because  it  is  used  far  more 
extensively  than  any  other  forms. 

If  the  two  portions  of  rope  which  sustain  the  weight  are 
inclined,  as  in  Fig.  86,  then  W  is  no  longer  equal  to  the  sum  of 
their  tensions,  as  it  is  in  the 
pulley,  but  is  always  less  than 
that,  according  to  the  follow- 
ing law  : 

The  power  is  to  the  weight 
as  the  sine  of  %  the  angle  is  to 
the  sine  of  the  whole  angle  be- 
tween the  parts  of  the  rope. 

Consider  the  point  E  as  in 
equilibrium  from  three  forces — 
P  along  E  F,  W  along  E  W  and 
a  force,  opposite  to  their  result- 
ant, along  E  B.  From  Art. 
44,  each  force  is  proportional 
to  the  sine  of  the  angle  between  the  other  two. 


Hence, 


P:  W  ::  (sin  WEB=)  sin  £  ED:  (sin  BEF=]  2  B  ED. 


123.  Change  in  the  Ratio  of  Power  and  Weight. — 
If  P  is  given,  all  the  possible  values  of  W  are  included  between 
W  -  0,  and  W  =  2  P. 

When  the  rope  is  straight  from  A  to  B  (Fig.  87),  so  that 
C  D  —  0,  then,  by  the  above  proportion,  W  =  0.  As  W  is 
increased  from  zero,  the  point 
G  descends  ;  and  when  D  C  = 
%  B  C,  then,  by  the  proportion, 
W  -  P.  In  that  case  D  C  B 
=  60°,  and  the  angles  A  C  B, 


FIG.  87. 


A  C  W,  and  B  C  W  are  equal 
(each  being  120°),  as  they  should 
be,  because  each  of  the  equal 
forces,  P,  P,  and  W,  is  as  the 
sine  of  the  angle  between  the  directions  of  the  other  two. 

But  when  W  has  increased  to  2  P,  it  descends  to  an  infinite 
distance  ;  for  then,  by  the  proportion,  G  D  =  B  C,  that  is,  the  side 
of  a  right-angled  triangle  is  equal  to  the  hypothenuse.  Thus,  the 
extreme  values  of  W  are  0  and  2  P. 

It  appears  from  the  foregoing  that  a  perfectly  flexible  rope 
having  weight  cannot  be  drawn  into  a  straight  horizontal  line  by 


MECHANICS. 


any  force,  however  great ;  for  C  cannot  coincide  with  D,  except 
when  W  =  0. 


FIG. 


124.  The  Branching  Rope. — When  C,  where  the  weight  is 
suspended,  is  a.  fixed  point  of  the  rope, 
we  have  a   branching   rope,  and   the 
principle  of  transmitted  tension  does 
not  apply  beyond  the  point  of  division. 

Let  P,  P',  and  W  (Fig.  88),  be 
given,  and  G  a  fixed  point  of  the  rope. 
Produce  W  G,  and  let  A  E,  drawn 
parallel  to  G  B,  intersect  it  in  R  The 
.sides  of  A  G  E  are  proportional  to  the 
given  forces  ;  therefore  its  angles  can 
be  found,  and  the  inclinations  of  A  G 
and  B  G  to  the  vertical  G  T7are  known. 


V.  THE  INCLINED  PLANE. 

125.  Relation  of  Power,  Weight,  and  Pressure  on 
the  Plane. — The  mechanical  efficiency  of  the  inclined  plane  is 
explained  on  the  principle  of  oblique  action  ;  that  is,  it  enables  us 
to  apply  the  power  to  balance  or  overcome  only  one  component  of 
the  weight,  instead  of  the  whole.  Let  the  weight  of  the  body  D, 
lying  on  the  inclined  plane  A  C  (Fig.  89),  be  represented  by  W; 
«nd  resolve  it  into  P" parallel,  and  .ZV  perpendicular  to  the  plane. 
N  represents  the  perpendicular  pressure,  and  is  equal  to  the 
reaction  of  the  plane  ;  F  is  the  force  by  which  the  body  tends  to 
move  down  the  plane. 

Let  a  —  the  angle   G,  the  in-  FIG.  89. 

clination  of  the  plane  ;  therefore 
W  J)  N  =  a.  Then  F  —  W.  sin  ; 
a  and  N  =  W.  cos  a. 

Now  suppose  a  force  P,  in  the 
direction  G  A,  applied  at  D, 
teeps  the  body  at  rest.  Evi- 
dently 


P  =  W  sin  a  =  W 


AS 
A~C' 


Hence,  with  a  force  acting  parallel  to  the  inclined  plane,  there  is 
equilibrium  when 

The  power  is  to  the  weight  as  the  height  to  the  length  of  the  in- 
clined plane. 


THE    INCLINED    PLANE. 


81 


FIG.  90. 


If  a  force  P  (Fig.  90),  having 
any  direction  whatever,  keeps 
the  body  at  rest,  then  the  re- 
sultant N  of  P  and  W  must  be 
perpendicular  to  the  plane,  for 
the  resultant  must  have  a  zero 
component  along  any  direction 
which  the  weight  could  move, 
whence  P  :  W : :  (sin  GNP  =  ) 
sin  a  :  sin  P  G  N. 

When  the  force  acts  in  line  parallel  to  the  base  (P  G  N  =  90° 
—  a),  we  have 

P  I  W  —  sin  a  :  cos  a  =  A  B  :  B  C,  or 

The  power  is  to  the  weight  as  the  height  is  to  the  base  of  the  in- 
clined plane. 

126.  Power  most  Efficient  when  Acting  Parallel  to  the 
Plane. — From  the  proportion  above 

P  .  sin  P  G  N 

]/\     ^^Z    ; * 


Now,  as  P  and  sin  a  are  given,  W  varies  as  sin  P  G  N,  which 
is  the  greatest  possible  when  P  G  N  =  90°  ;  that  is,  when  the 
power  acts  in  a,  line  parallel  to  the  plane. 

Whether  the  angle  P  G  N  diminishes  or  increases  from  90°, 
its  sine  diminishes,  and  becomes  zero,  when  P  G  N  =  0°,  or  180°. 
Therefore  W  —  0,  or  no  weight  can  be  sustained,  when  the  power 
acts  in  the  line  G  N,  perpendicular  to  the  plane,  either  toward  the 
plane  or  from  it. 

127.  Expression  for  Perpendicular  Pressure.  —  From  the 
triangle  P  G  N  we  obtain 

N  :  W  :  :  sin  G  P  N  :  sin  P  G  N  :  :  sin  P  G  W  :  sin  P  G  N; 
8in  P  G  W 


If  the  power  acts  in  a  line  parallel  to  the  inclined  plane, 
P  G  W  =  90°  +  a,  P  G  N  =  90°,  and  N  =  W  *™  ^Itf  ^  = 
W  cos  a. 

If  the  power  acts  in  a  line  parallel  to  the  base  of  the  inclined 

plane,  P  G  W  =  90°,  P  G  N  =  90°  -  a,  and  N  =  W—  = 

cos  a 
JFseca. 


82  MECHANICS. 

If  the  power  acts  in  a  line  perpendicular  to  the  inclined  plane, 
P  GW  =  a,  PG  N=0°,  and  JV  =  JF-  -  =  <x  . 


FIG.  91. 


128.  Equilibrium  between  Two  Inclined  Planes. — If  a 
body  rests,  as  represented  in  Fig.  91,  between  two  inclined  planes, 
the  three  forces  which  retain  it  are  its  weight,  and  the  resistances 
of  the  planes.  Draw  H  F  and  L  F 
perpendicular  to  the  planes  through 
the  points  of  contact,  and  G  F  verti- 
cally through  the  centre  of  gravity  of 
the  body.  Since  the  body  is  in  equi- 
librium, these  three  lines  will  pass 
through  the  same  point  (Art.  43).  Let 
that  point  be  F,  and  draw  G  P  paral- 
lel to  L  F,  and  M  K  parallel  to  the 
horizon.  G  P  Fis,  similar  to  K  C M. 
Therefore  (since  Pressure  on  A  C  :  Pr. 
on  DC1::  P  G  :  F  P), 


Pressure  on  A  C  :  Pr.  on  D  C  : 


KC-.MC, 

sin  M  :  sin  K, 

sin  D  C  E  :  sin  A  C  B. 


That  is,  when  a  body  rests  between  two  planes,  it  exerts  pressures 
on  them  which  are  inversely  as  the  sines  of  their  inclinations  to 
the  horizon. 

If,  therefore,  one  of  the  planes  is  horizontal,  none  of  the 
pressure  can  be  exerted  on  any  other  plane.  It  is  friction  alone 
which  renders  it  possible  for  a  body  on  a  horizontal  surface  to 
lean  against  a  vertical  wall. 

129.  Bodies  Balanced  on  Two  Planes  by  a  Cord 
passing  over  the  Ridge.— Let  P  and  W  balance  each  other  on 
the  planes  A  D  and  A  C  (Fig.  92), 
which  have  the  common  height  A  B, 
"by  means  of  a  cord  passing  over  the 
fixed  pulley  A.  The  tension  of  the 
cord  is  the  common  power  which  pre- 
vents each  body  from  descending ;  and 
as  the  cord  is  parallel  to  each  plane,  c 
we  have  (calling  the  tension  t), 

t  :  P  : :  A  B  :  A  D ; 

and£:  W ::  A  B  :  A  C; 

/.  P:  Wr.A  D:AC; 


FIG.  92. 


THE    SCREW.  S3 

that  is,  the  forces,  in  case  of  equilibrium,  are  directly  as  the  lengths 
of  the  planes. 

130.  Questions  on  the  Inclined  Plane. — 

1.  If  a  horse  is  able  to  raise  a  weight  of  440  Ibs.  perpendicu- 
larly, what  weight  can  he  raise  on  a  railway  having  a  slope  of  five 
degrees?  Ans.  5048.5  Ibs. 

2.  The  grade  of  a  railroad  is  20  feet  in  a  mile  ;  what  force  must 
be  exerted  to  sustain  any  given  weight  upon  it  ? 

Ans.  I  Ib.  for  every  264  Ibs. 

3.  What  force  is  requisite  to  hold  a  body  on  an  inclined  plane, 
by  pressing  perpendicularly  against  the  plane  ? 

Ans.  An  infinite  force. 

4.  A  certain  force  was  able  to  sustain  500  tons  on  a  plane  of 
7£°  ;  but  on  another  plane  it  could  sustain  only  400  tons  ;  what 
was  the  inclination  of  the  latter?  Ans.  9°  23'  25".    ' 

5.  Equilibrium  on  an  inclined    plane   is  produced   when  the 
force,  weight,  and  perpendicular  pressure  are,  respectively,  9,  13, 
and  6  Ibs.  ;   what  is  the  inclination  of   the  plane,  and  what   angle 
does  the  force  make  with  the  plane  ? 

Ans.  'a  —  37°  21'  26".      Inclination     of   force    to    plane 
=  28°  46'  54". 

6.  A  force  of  10  Ibs.,  acting  parallel  to  the  plane,  supports  a 
certain  weight;  but  it  requires  a  force  of  12  Ibs.  parallel  to  the  base 
to  support  it.     What  is  the  weight  of  the  body,  and  what  is  the 
inclination  of  the  plane  ? 

Ans.  W  =  18.09  Ibs.     a  =  33°  33'  25". 

7.  To  support  a  weight  of  500  Ibs.  upon  an  inclined  plane  of 
50°  inclination   to   the   horizon,  a  lifting   force  is  applied   whose 
direction  makes  an  angle  of  75°  with   the  horizon.     What   is   the 
magnitude  of  this  force,  and  the  pressure  of  the  weight  against 
the  plane  ?  Ans.  P  =  422.6  Ibs.     N  =  142.8  Ibs. 

VI  THE  SCREW. 

131.  Reducible   to   the    Inclined   Plane. — The  screw  is  a 
cylinder  having  a  spiral  ridge  or  thread  around  it,  which  cuts  at  a 
constant  oblique  angle  all  the  lines  of  the  surface  parallel  to   the 
axis   of  the  cylinder.     A  hollow  cylinder,  called  a  imt,  having  a 
similar  spiral  within  it,  is  fitted  to  move  freely  upon  the  thread  of 
the  solid  cylinder.     In  Fig.  93,  let  the  base  A  B  of  the  inclined 
plane  A   C  be  equal  to  twice  the  circumference  of  the  cylinder 
A'  E ;  then  let  the  plane  be  wrapped  about  the  cylinder,  bringing 
the  points  A,  F,  and  B,  to  the  point  A' ;  then  will  A  G  describe 
two  revolutions  of  the  thread  from  A'  to  C'.     Therefore  the  me- 


MECHANICS. 


chanical  relations  of  the  screw  are  the  same  as  of  the  inclined 

plane. 

FIG.  93. 


If  a  weight  be  laid  on  the  thread  of  the  screw,  and  a  force  be 
applied  to  it  horizontally  in  the  direction  of  a  tangent  to  the 
cylinder,  the  case  is  exactly  analogous  to  that  of  a  body  moved  on 
an  inclined  plane  by  a  force  parallel  to  the  base.  Let  r  be  the 
radius  of  the  cylinder,  then  2  TT  r  is  the  circumference ;  also  let  d 
be  the  distance  between  the .  threads,  (that  is,  from  any  point  of 
one  revolution  to  the  corresponding  point  of  the  next,)  measured 
parallel  to  the  axis  of  the  cylinder ;  then  2  n  r  is  the  base  of  an  in- 
clined plane,  and  d  its  height.  Therefore  (Art.  125), 
P  :  Wr.d  :2nr;  or, 

The  power  is  to  the  weight  as  the  distance  between  the  threads 
measured  parallel  to  the  axis,  is  to  the  circumference  of  the  screw. 

If  instead  of  moving  the  weight  on  the  thread  of  the  screw,  the 
force  is  employed  to  turn  the  screw  itself,  while  the  weight  is  free 
to  move  in  a  vertical  direction,  the  law  is  the  same.  Thus, 
whether  the  screw  A'  E  is  allowed  to  rise  and  fall  in  the  fixed  nut 
G  H,  or  whether  the  nut  rises  and  falls  on  the  thread  of  the  screw, 
while  the  latter  is  revolved,  without  moving  longitudinally,  in 
each  case,  P  :  W : :  d  :  2  TT  r. 

132.  The  Screw  and  Lever  Combined.— The  screw  is  so 
generally  combined  with  the  lever  in  practical  mechanics,  that  it 
is  important  to  present  the  law  of  the 
compound  machine.  Let  A  F  (Fig.  94) 
be  the  section  of  a  screw,  and  suppose 
B  C,  a  lever  of  the  second  order,  to  be 
applied  to  turn  it.  The  fulcrum  is  at  (7, 
the  power  acts  at  B,  and  the  effect  pro- 
duced by  the  lever  is  at  A,  the  surface  of 
the  cylinder.  Call  that  effect  x,  and  let 
d  =  the  distance  between  the  threads; 
then, 


Fm.  94. 


THE    ENDLESS     SCREW.  85 

P  :x  ::  A  C :  B  G, 
and  x  :  W::d:2rf  A  C; 
compounding  and  reducing,  we  have 

P  :  W  ::  d:  2  TT  B  C;  that  is, 

The  poiver  is  to  the  weight  as  the  distance  between  the  threads, 
•measured  parallel  to  the  axis,  to  the  circumference  described  by  the 
power. 

The  law  as  thus  stated  is  applicable  to  the  screw  when  used 
with  the  lever  or  without  it. 

133.  The  Endless  Screw. — The  screw  is  so  called,  when 
its  thread  moves  between  the  teeth  of  a  wheel,  thus  causing  it  to 
revolve.    It  is  much  used  for  diminish- 
ing very  greatly  the  velocity  of  the 

weight. 

Let  P  Q  (Fig.  95)  be  the  radius  of 
the  crank  to  which  the  power  is  ap- 
plied; d,  the  distance  between  the 
threads ;  R,  the  radius  of  the  wheel ; 
r,  the  radius  of  the  axle;  and  call  the 
force  exerted  by  the  thread  upon  the 
teeth,  x.  Then, 

P:x::d:2n  x  P  Q, 
andz  :  W  ::r  :  R', 

.-.  P  :  W  : :  dr  :2  n  x  R  x  P  Q. 

If,  for  example,  P  Q [  =  30  inches,  d=\  in.,  R  =  18  in. ; 
r  -f  t  =  2  in. ;  then  W  '=  1696  P,  and  moves  with  1696  times 
less  velocity  than  P. 

134.  The  Right  and  Left  Hand  Screw.— The  common 

iorm  of  screw  is  called  the  right-hand  screw,  and  may  be  described 
thus  :  if  the  thread  in  its  progress  along  the  length  of  the  cylinder, 
passes  from  the  left  over  to  the  right,  it  is  called  a  right-hand  screw. 
Hence,  a  person  in  driving  a  screw  forward  turns  it  from  his  left 
over  (not  under)  to  his  right,  and  in  drawing  it  back  he  reverses 
this  movement.  Fig.  93  represents  a  right-hand  screw. 

The  left-hand  screw  is  one  whose  thread  is -coiled  in  the  oppo- 
site direction, — that  is,  it  advances  by  passing  from  right  over  to 
left.  This  kind  is  used  only  when  there  is  special  reason  for  it. 
For  example,  the  screws  which  are  cut  upon  the  left-hand  ends  of 
carriage  axles  are  left-hand  screws;  otherwise  there  would  be 
•danger  that  the  friction  of  the  hub  against  the  nut  might  turn 
the  nut  off  from  the  axle.  Also,  when  two  pipes  for  conveying 
gas  or  steam  are  to  be  drawn  together  by  a  nut,  one  must  have  a 
right-hand,  and  the  other  a  left-hand  screw. 


MECHANICS. 


135.  Questions  on  the  Screw. — 

1.  The  distance  between  the  threads  of  a  screw  is  one  inch, 
the  bar  is  two  feet  long  from  the  axis,  and  the  power  is  30  Ibs. ; 
what  is  the  weight  or  pressure  ?  Ans.   4523.89  Ibs. 

2.  The  bar  is  three  feet  long,  reckoned  from  the  axis,  P  =  60 
Ibs.,  W  —  2240  Ibs. ;   what  is  the  distance  between  the  threads? 

Ans.  6.058  inches. 

3.  A  compound  machine  consists  of  a  crank,  an  endless  screw, 
a  wheel  and  axle,  a  pulley,  and  an  inclined  plane.     The  radius  of 
the  crank  is  18  inches  ;.  the  distance  between  the  threads  of  the 
screw,  one  inch  ;  the  radius  of  the  wheel  on  which  the  screw  acts, 
two  feet ;  the  radius  of  the  axle,  6  inches ;  the  pulley  block  has- 
two  movable  pulleys  with  one  rope,  the  power  exerted  by  the 
pulley  being  parallel  to  the  plane,  and  the  inclination  of  the  plane 
to  the  horizon  is  30°.     What  weight  on  the  plane  will  be  balanced 
by  a  power  of  100  Ibs.  applied  to  the  crank  ? 

Ans.  361911.474  Ibs. 

VII.  THE  WEDGE. 

136.  Definition   of  the  Wedge,  and    the    Mode    of 
Using. — The  usual  form  of  the  wedge  is  a.  triangular  prism,  two- 
of  whose  sides  meet  at  a  very  acute  angle.     This  machine  is  used 
to  raise  a  weight  by  being  driven  as  an  inclined  plane  underneath 
it,  or  to  separate  the  parts  of  a  body  by  being  driven  between 
them.     When  it  is  used  by  itself,  and  does  not  form  part  of  a 
compound  machine,  force  is  usually  applied  by  a  blow,  which  pro- 
duces an  intense  pressure  for  a  short  time,  sufficient  to  overcome 
a  great  resistance. 

137.  Law  of  Equilibrium. — Whatever  be 
the  direction  of  the  blow  or  force,  we  may  sup- 
pose it  to  be  resolved  into  two  components,  one 
perpendicular  to  the  back  of  the  wedge,  and  the 
other  parallel  to  it.    The  latter  produces  no  effect. 
The  same  is  true  of  the  resistances  ;  we  need  to 
consider  only  those  components  of  them  which 
are  perpendicular  to  the  sides  of  the  wedge. 

Let  M NO  (Fig.  96)  represent  a  section  of 
the  wedge  perpendicular  to  its  faces  ;  then  P  A, 
Q  A,  and  R  A,  drawn  perpendicular  to  the  faces 
severally,  show  the  directions  of  the  forces  which 
hold  the  wedge  in  equilibrium.  Taking  A  B  to 
represent  the  power,  draw  B  C  parallel  to  R  A, 
and  we  have  the  triangle  A  B  C,  whose  sides 
represent  these  forces.  But  A  B  C  is  similar  to  M  N  0, 


their 


THE    KNEE-JOINT.  87 

.sides  are  respectively  perpendicular  to  each  other.     Hence,  calling 
the  forces  P,  Q,  and  R,  respectively, 

P:Q  ::MN:MO: 
and  P:R-.'.MN:NO; 
that  is,  there  is  equilibrium  in  a  wedge,  when 

The  power  is  to  the  resistances  as  the  back  of  the  tvedge  to  the 
sides  on  which  the  resistances  respectively  act. 

If  the  triangle  is  isosceles,  the  two  resistances  are  equal,  as  the 
proportions  show  ;  and  P  is  to  either  resistance,  R,  as  the  breadth 
of  the  back  to  the  length  of  the  side. 

If  the  resisting  surfaces  touch  the  sides  of  the  Avedge  only  in 
one  point  each,  then  Q  A  and  R  A,  drawn  through  the  points  of 
contact,  must  meet  A  P  in  the  same  point  (Art.  43)  ;  otherwise 
the  wedge  will  roll,  till  one  face  rests  against  the  resisting  body 
in  two  or  more  points. 

The  efficiency  of  the  wedge  is  usually  very  much  increased  by 
•combining  its  own  action  with  that  of  the  lever,  since  the  point 
where  it  acts  generally  lies  at  a  distance  from  the  point  where  the 
effect  is  to  be  produced.  Thus,  in  splitting  a  log  of  wood,  the 
resistance  to  be  overcome  is  the  cohesion  of  the  fibers  ;  and  this 
force  is  exerted  at  a  distance  from  the  wedge,  while  the  fulcrum 
is  a  little  further  forward  in  the  solid  wood. 

VIII.  THE  KNEE-JOINT. 

138.  Description  and  Law  of  Equilibrium. — The  knee- 
joint  consists  of  two  bars,  usually  equal,  hinged  together  at  one 
-end,  while  the  others  are  at  liberty  to  separate  in  a  straight  line. 
The  power  is  applied  at  the  hinge,  tending  to  thrust  the  bars 
into  a  straight  line ;  the  weight  is  the  force  which  opposes  the 
separation. 

FIG.  97. 


Suppose  that  A  B  and  A  D  (Fig.  97)  are  equal  bars,  hinged 
together  at  A  ;  and  that  the  bar  A  B  is  free  only  to  revolve  about 


88 


MECHANICS. 


the  axis  B,  while  the  end  D  of  the  other  bar  can  move  parallel  to 
the  base  E F.  UP  urges  A  toward  the  base,  it  tends  to  move  Z> 
further  from  the  fixed  point  B.  The  force  P',  which  opposes 
that  motion,  is  represented  in  the  figure  by  the  weight  W.  The 
law  of  equilibrium  is, 

The  power  is  to  the  weight  as  twice  the  height  of  the  joint  to 
half  the  distance  between  the  ends  of  the  bars. 


Eesolving  the  force  P  in  the  direction  of  A  B  and  A  D,  we 
have,  Fig.  98, 

P  :  T::AH:A  G  ::  2  A  C  :  A  D, 

in  which  T  stands  for  the  component  of  P  in  the  direction  A  G> 
called  the  thrust. 

This  component  T  acts  at  D  and  must  be  again  resolved  in- 
the  directions  D  L  and  D  M,  of  which  D  L  is  equal  and  opposed 
to  W,  and  D  M  is  equal  and  opposed  to  the  upward  resistance  of 
the  plane  on  which  the  block  D  slides,  giving  the  proportion 

T'.W'.'.DK-.D  L  or  M K '.:  A  D  :  C D. 

Multiplying  like  terms  of  the  two  proportions  and  omitting  com- 
mon factors,  we  have, 

P:  W::2  A  C:  CD. 

139.   Ratio   of   Power   and  Weight  Variable. — It  is- 

obvious  that  the  ratio  between  power  and  weight  is  different  for 
different  positions  of  the  bars.  As  A  is  raised  higher  CD  dimin- 
ishes, and  when  the  bars  are  parallel,  we  have 

P:  W::2AC:Q', 


VIRTUAL    VELOCITIES.  89 

that  is  to  say,  the  power  has  no  efficiency.  But  as  A  approaches 
the  base  A  C  diminishes,  and  at  last  we  have,  when  B  A  and 
A  D  are  in  the  same  line, 

P:  JF::0:  B  A. 

Hence  the  weight  or  resistance  in  such  case  is  infinite  as  com- 
pared with  the  power  applied.  The  indefinite  increase  of  effi- 
ciency in  the  power,  which  occurs  during  a  single  movement, 
renders  this  machine  one  of  the  most  useful  for  many  purposes, 
as  printing  arid  coining. 

Questions  on  the  knee-joint. — 

1.  A  power  of  50  Ibs.  is  exerted  on  the  joint  A  (Fig.  97)  ; 
compare  the  weight  which  will  balance  it,  when  B  A  D  is,  90°, 
and  when  it  is  160°.  Ans.  25  Ibs.  and  141.78  Ibs. 

2.  When  the  angle  between  the  bars  is  110°,  a  certain  power 
just  overcomes  a  weight  of  65  Ibs.;  what  must  be  the  angle,  in 
order  that  the  weight  overcome  may  be  five  times  as  great  ? 

Ans.  164°  3'  22". 

PRINCIPLE  OF  VIRTUAL  VELOCITIES. 

140.  Definition. — The  virtual  velocity  of  a  point,  with  respect 
to  any  force,  is  the  product  of  its  actual  velocity  by  the  cosine  of 
the  angle  which  its  actual  path  makes  with  the  direction  of  the 
force.     Thus,  let  a  point  A  (Fig.  99)  be  acted  upon  by  a  force 
P  in  the  direction  A  c,  and  be- 
cause of  some  other  external  force 

or  resistance  suppose  the  point  ^4 

to  be  constrained  to  move  in  the      ^^^ 

line  A  A'  to  A'  in  any  unit  of     A  a  c  >F 

time  :  then  A  d,  the  projection 

of  A  A'  upon  A  c,  is  the  virtual 

velocity  of  the  point  A  with  reference  to  the  force  P. 

141.  The  Point  of  Application  Moving  in  the  Line  of 
the  Force. — It  can  be  shown,  in  every  case,  that  the  velocities, 
when  reckoned  in  the  direction  in  which  the  forces  act.,  are  inversely 
as  the  forces. 

Some  examples  are  first  given  in  which  the  point  of  applica- 
tion moves  in  the  line  in  which  the  force  acts. 

In  the  straight  lever  (Fig.  100),  which  is  in  equilibrium  by  the 

weights  P  and  W,  suppose  a 

FIG.  100.  slight  motion  to  exist ;  then 

the  velocity  of  each  will  be 
as  the  arc  described  in  the 
same  time  ;  but  the  arcs  are 
similar,  since  they  subtend 


90  MECHANICS. 

equal  angles.  Therefore,  if  V  =  velocity  of  P,  and  v  =  Telocity 
of  W. 

V:v::AP:£  W::A  G\B  C ; 
but  it  has  been  shown  (Art.  106)  that 

P:  W::S  C  :  A  <7; 
.-.    F  :  v  ::    W   :   P; 

that  is,  the  velocity  of  the  power  is  to  the  velocity  of  the  weight 
as  the  weight  to  the  power.  Hence,  P  x  its  velocity  =  W  x  its 
velocity ;  that  is,  the  momentum  of  the  power  equals  the  mo- 
mentum of  the  weight. 

In  the  wheel  and  axle,  let  R  and  r  be  the  radii,  and  suppose 
the  machine  to  be  revolved ;  then  while  P  descends  a  distance 
equal  to  the  circumference  of  the  wheel  =  2  n  R,  the  weight 
ascends  a  distance  equal  to  the  circumference  of  the  axle  =  2  n  r. 
Therefore, 

V  :v::2  n  R  :2nr::  R  :  r; 
but  (Art.  113),  P  :  W : :  r  :  R ; 

.-.  V:v::  W  :  P ; 

or,  the  velocities  are  inversely  as  the  weights  ;  and  P  x  F==  Wx  v, 
the  momentum  of  the  power  equals  the  momentum  of  the 
weight. 

In  the  fixed  pulley  the  velocities  are  obviously  equal ;  and  we 
have  before  seen  that  the  power  and  weight  are  equal;  therefore 
the  proportion  holds  true,  F  :  v  : :  W  :  P ;  and  the  momenta  are 
equal. 

In  the  movable  pulley,  if  n  is  the  number  of  sustaining  parts 
of  the  cord,  when  W  rises  any  distance  =  x,  each  portion  of  cord 
is  shortened  by  the  distance  x,  and  all  these  n  portions  pass  over 
to  P,  which  therefore  descends  a  distance  =  n  x, 

Hence, 

V  :  v  : :  n  x  :  x  : :  n  :  1 ; 
but  (Art.  120),  P  :  W ::  1  :  «; 

.".  V:v::  W:  P; 
as  in  all  the  preceding  cases. 

In  the  screw  (Fig.  94),  while  the  power  describes  the  circum- 
ference =  2  TT  x  B  C,  the  weight  moves  only  the  distance  =  «?; 
therefore, 

V:  v::27T  x  B  C:d; 
but  (Art.  131),          P:  W-.-.d-.Z*  x  B  C '; 
.'.  F:  v::  W:  P; 

therefore  the  momentum  of  the  power  equals  the  momentum  of 
the  weight,  as  before. 


VIRTUAL    VELOCITIES. 


91 


142.  The  Point  of  Application  Moving  in  a  Different 
Line  from  that  in  which  the  Force  Acts. — The  cases  thus 
far  noticed  are  the  most  obvious  ones,  because  the  points  of  appli- 
cation of  force  and  weight  actually  move  in  the  directions  in  which 
their  force  is  exerted.  The  case  of  the  inclined  plane  will  illus- 
trate the  principle,  when  the  point  of  application  does  not  move 
in  the  direction  of  the  force. 

First,  let  P  (Fig.  101)  act  parallel  to  the  plane,  and  suppose 
the  body  to  be  moved  either  up  or  down  the  plane  a  distance  equal 
to  G  d.  That  is  the  velocity 

of  the  force.     But  in  the  di-  FIG.  101. 

rection  of  the  weight  (force  of 
gravity)  the  body  moves  only 
the  distance  6  d.  Therefore 
the  velocity  of  the  force  is  to 
the  velocity  of  the  weight 
(each  being  reckoned  in  the 
line  of  its  action)  as  G  d  to 
bd. 

By  similar  triangles,     G  d  :  b  d    :  A  C  :  A  B  ; 
or     V :    v     \AC\AB. 

But  (Art.  125),  P  :    W   :  A  B  :  A  C; 

.-.   V\    v     :    W    :P. 

Again,  let  the  force  act  in  any  oblique  direction,  as  G  e.  If 
the  body  moves  over  G  d,  draw  d  e  perpendicular  to  G  e  ;  then  G  e 
is  the  distance  passed  over  in  the  direction  of  the  force,  and  b  d  in 
the  direction  of  the  weight.  Assuming  the  unit  of  time  to  be  con- 
sumed in  the  motion,  these  lines  represent  the  velocities  and  we 
have 

G  e    b  d        .  _  .,     v  •'•  .  W  ^». «» 

V :  v  =  YTj  '•  ~7Tj  —  (cos  e  G  d  =)  sin  P  G  N :  sm  a. 


But  (Art.  126), 


P  :  W  =  sin  a  :  sin  P  G  N; 
V:   v  =  W :  P. 


Consider  in  the  foregoing  that  the  machines  act  for  a  unit  time. 
Then  V  and  v  represent  the  distances  through  which  the  force  and 
resistance  have  moved.  P  V  represents  the  work  done  by  the 
force  ;  W  u  the  work  performed  upon  the  resisting  body.  These 
are  always  equal,  and  hence  a  machine  does  not  give  out  more 
work  than  is  impressed  upon  it.  However,  the  resisting  force  may 
be  of  any  desired  magnitude,  and  the  machine  will  enable  a  smaller 
force  to  produce  a  given  motion  against  it,  by  itself  moving  a 
greater  distance.  On  the  other  hand,  by  a  machine,  a  large  force 
with  small  velocity  can  be  made  to  produce  great  velocity  against  a 
small  resistance. 


92  MECHANICS. 

FRICTION  IN  MACHINERY. 

143.  The  Power  and  Weight  not  the  only  Forces  in 
a  Machine. — For  each  machine  a  certain  proportion  has  been 
given,  which  insures  equilibrium.     And  it  is  implied  that  if  either 
the  power  or  the  weight  be  altered,  the  equilibrium  will  be  de- 
stroyed.    But  practically  this  is  not  true  ;    the  power  or  weight 
may  be  considerably  changed,  or  possibly  one  of  them  may  be 
entirely  removed,  and  the  machine  still  remain  at  rest.     The  ob- 
struction which  prevents  motion  in  such  cases,  and  which  always 
exists  in  a  greater  or  less  degree,  arises  from  friction ;  and  fric- 
tion is  caused  by  roughness  in  the  surfaces  which  rub  against 
each  other.     The  minute  elevations  of  one  surface  fall  in  between 
those  of  the  other,   and   directly  interfere  with   the   motion   of 
either,  while  they  remain  in  contact.     Polishing  diminishes  the 
friction,  but  can  never  remove  it,  for  it  never  removes  all  rough- 
ness. 

The  coefficient  of  friction  is  the  fraction  whose  numerator  is  the 
force  required  to  overcome  the  friction,  and  its  denominator  the 
normal  pressure  between  the  bodies. 

Let  fi  =  coefficient ;  Pn  —  normal  pressure ;  F  =  force  required 
to  overcome  friction.  Then 

F^  pPn.     .-.  p.=  F/Pn. 

As  friction  always  tends  to  prevent  motion,  and  never  to  pro- 
duce it,  it  is  called  a  passive  force.  It  assists  the  power,  when  the 
weight  is  to  be  kept  at  rest,  but  opposes  it,  when  the  weight  is  to- 
be  moved.  There  are  other  passive  forces  to  be  considered  in  the 
study  of  science,  but  no  other  has  so  much  influence  in  the  oper- 
ations of  machinery  as  friction. 

144.  Modes  of  Experimenting. — When  one  surface  slides 
on  another,  the  friction  which  exists  is  called  the  sliding  friction  ; 
but  when  a  wheel  rolls  along  a  surface,  the  friction  is  called  rolling 
friction.     The  sliding  friction  occurs  much  more  in  machines  than 
the  rolling  friction. 

Experiments  for  ascertaining  the  laws  of  friction  may  be  per- 
formed by  placing  on  a  table  a  FlG  103 
block  of  three  different  dimen- 
sions, and  measuring  its  friction 
under    different    circumstances 
by  weights  acting  on  the  block 
by  means  of  a  cord  and  pulley, 
as  represented  in  Fig.  102.     This 
was  the  method  by  which  Coulomb  first  ascertained  the  laws  of 
friction. 


FRICTION    IN    MACHINERY.  93, 

Another  mode  is  to  place  the  block  on  an  inclined  plane,  whose 
angle  can  be  varied,  and  then  find  the  relative  friction  in  different 
cases,  by  the  largest  inclination  at  which  it  will  prevent  the  block 
from  sliding.  For,  when  Won  the  inclined  plane  A  B  (Fig.  103) 
is  on  the  point  of  sliding  down, 
friction  is  the  power  which,  acting 
parallel  to  the  plane,  is  in  equili- 
brium with  that  component  of  the 
weight  which  tends  to  move  the 
block  down  the  plane. 

This  component  parallel  to  the 
plane,  is  TFsin  A,  and  the  normal 
pressure  W  cos  A  ;  hence,  calling  the  coefficient  of  friction  u,  we 

have  (Art.  143)  u  =  -rr- j  =  tan  A,  or  the  coefficient  of  fric- 

\V  COS  -A. 

lion  is  equal  to  the  tangent  of  the  angle  of  inclination  of  the  plane. 
For  example,  suppose  a  block  of  cast  iron  to  rest  upon  an  oak 
plank,  and  that  the  end  of  the  plank  is  raised  so  that  the  block 
slides  with  uniform  motion  down  the  plane  ;  then  the  angle  A 
will  be  found  by  actual  measurement  to  be  about  26°,  the  natural 
tangent  of  which  is  .48773  ;  hence,  in  pounds,  49  per  cent,  of  the 
weight  will  represent  the  force  in  pounds  required  to  overcome 
the  friction. 

145.  Laws  of  Sliding  Friction. — The  laws  of  sliding; 
friction  on  which  experimenters  are  generally  agreed  are  the 
following : 

1.  Friction  varies  as  the  pressure. — If  weights  are  put  upon, 
the  block,  it  is  found  that  a  double  weight  requires  a  double  force 
to  move  it,  a  triple  weight  a  triple  force,  &c. 

2.  It  is  the  same,  however  great  or  small  the  surface  on  which 
the  body  rests. — If  the  block  be  drawn,  first  on  its  broadest  side/ 
then  on  the  others  in  succession,  the  force  required  to  overcome 
friction  is  found  in  each  case  to  be  the  same.     Extremes  of  size 
are,  however,  to  be  excepted.     If  the  loaded  block  were  to  rest 
on  three  or  four  very  small  surfaces,  the  obstruction  might  be 
greatly  increased  by  the  indentations  thus  occasioned  in  the  sur- 
face beneath  them. 

3.  Friction  is  a  uniformly  retarding  force. — That  is,  it  destroys 
equal  amounts  of  motion  in  equal  times,  whatever  may  be  the 
velocity,  like  gravity  on  an  ascending  body. 

4.  Friction,  at  the  first  moment  of  contact  is  less  than  after  con- 
tact has  continued  for  a  time. — And  the  time  during  which  fric- 
tion increases,  varies  in  different  materials.     The  friction  of  wood 
on  wood  reaches  its  maximum  in  three  or  four  minutes  ;  of  metal 


94  MECHANICS. 

on  metal,  in  a  second  or  two  ;  of  metal  on  wood,  it  increases  for 
several  days.  As  any  jar  or  vibration  changes  at  once  the  friction 
of  rest  to  that  of  motion,  the  coefficients  to  be  considered  in 
determining  the  stability  of  any  structure  should  be  those  of 
motion. 

5.  Friction  is  less  between  substances  of  different  kinds  than  be- 
tween those  of  the  same  kind. — Hence,  in  watches,  steel  pivots  are 
made  to  revolve  in  sockets  of  brass  or  of  jewels,  rather  than  of 
steel. 

146.  Friction  of  Axes. — In   machinery,  the   most  common 
case  of  friction  is  that  of  an  axis  revolving  in  a  hollow  cylinder,  or 
the  reverse,  a  hollow  cylinder  revolving  on  an  axis.     These  are 
cases  of  sliding  friction,  in  which  the  force  that  overcomes  the 
friction,  usually  acts  at  the  circumference  of  a  wheel,  and  there- 
fore at  a   mechanical  advantage.     Thus,  the  friction  on  an  axis, 
whose  coefficient  is  as  high  as  20  per  cent.,  requires  a  force  of 
only  two  per  cent,  to  overcome  it,  provided  the  force  acts  at  the 
circumference  of  a  wheel  whose   diameter   is   ten   times  that  of 
the  axis. 

147.  Rolling  Friction. — This  form  of  friction  is  very  much 
less  than  the  sliding,  since  the  projecting  points  of  the  surfaces 
do  not  directly  encounter  each  other,  but  those  of  the  rolling  wheel 
are  lifted  up  from  among  those  of  the  other  surface,  as  the  wheel 
advances. 

By  the  use  of  the  apparatus  described  in  Art.  144,  the  laws  of 
the  rolling  are  found  to  be  the  same  as  those  of  the  sliding  friction. 
But  on  account  of  the  manner  in  which  this  form  of  friction  is  over- 
come, there  is  this  additional  law  : 

The  force  required  to  roll  the  wheel  varies  inversely  as  the  di- 
ameter. 

For  the  force,  acting  at  the  centre  of  the  wheel  to  turn  it  on 
its  lowest  point  as  a  momentary  fulcrum,  has  the  advantage  of 
greater  acting  distance  as  the  diameter  increases. 

It  is  the  rolling  friction  which  gives  value  to  friction  wheels, 
as  they  are  calleh.  When  it  is  desirable  that  a  wheel  should  re- 
volve with  the  least  possible  friction,  each  end  of  its  axis  is  made 
to  rest  in  the  angle  between  two  other  wheels  placed  side  by  side, 
as  shown  in  Fig.  104.  The  wheel  is  obstructed  only  by  the  rolling 
friction  on  the  surfaces  of  the  four  wheels,  and  the  retarding  effect 
of  the  sliding  friction  at  the  pivots  of  the  latter  is  greatly  reduced 
on  the  principle  of  the  wheel  and  axle. 

The  sliding  friction  is  diminished  by  lubricating  the  surface, 
the  rolling  fiiction  is  not. 


LIMITING    ANGLE    OF    FRICTION.  95 

FIG.  104. 


148.  Advantages   of  Friction. — Friction  in  machinery  is 
generally  regarded  as  an  evil,  since  more  force  is  on  this  account 
required  to  do  the  work  for  which  the  machine  is  made.     But  it 
is  easy  to  see  that,  in  general,  friction  is  of  incalculable  value,  or 
rather,  that  nothing  could  be  accomplished  without  it.     Objects 
stand  firmly  in  their  places  by  friction  ;  and  the  heavier  they  are, 
the  more  firmly  they  stand,  because  friction  increases  with  the 
pressure.     All  fastening  by  nails,  bolts,  and  screws,  is  due  to  fric- 
tion.    The  fibres  of  cotton,  wool,  or  silk,  when  intertwined  with 
each  other,  form  strong  threads  or  cords,  only  because  of  the  power 
of  friction.     Without  friction   it  would  be  impossible  to  walk,  or 
even  to  stand,  or  to  hold  anything  by  grasping  it  with  the  hand. 

149.  Limiting  Angle  of  Friction. — If  two  surfaces  are  in 
contact  and  forces  be  applied,  oblique  to  the  surface  of  contact, 
no  motion  will  result  so  long  as  the  resultant  of  all  forces  makes 
with  the  normal  an  angle  whose  tangent  is  less  than  the  coefficient 
of  friction,  no  matter  how  great  this  resultant  force  may  be. 

Thus  (Fig.  105)  let  a  block  of  iron  rest  upon  a  surface  of  oak, 
as  in  the  case  heretofore  considered,  and  let 
the  force  P  be  applied.     In   this  case  the  FIG.  105. 

forces  are  P  and  W,  and  their  resultant  is  R 
(or  K)  which  may  be  considered  in  their 
stead.  The  component  of  E\  which  tends  to 
produce  motion,  is  E'  sine  a,  and  the  total 

.  _,  E'  sine  a 

normal  pressure  is  R  cos  a.     If  -=-, —   —  = 

R  cos  a 

tan  a,  is  less  than  the  coefficient  of  fi'iction  no 
motion  can  result ;  that  is  to  say,  if  a  is  less 


<)6  MECHANICS. 

than  the  inclination  of  the  plane,  in  Art.  144,  there  will  be  stable 
equilibrium. 

The  greatest  angle  a  which  the  resultant  of  all  the  forces  can 
make  with  the  normal  without  producing  sliding  motion  of  the 
surfaces  is  called  the  limiting  angle  of  f notion,  and  its  tangent  is 
equal  to  the  coefficient  of  friction. 

PKOBLEMS  ON  FRICTION. 

1.  A  force  of  6  kilos  is  just  sufficient  to  move  a  body  weighing 
48  kilos   uniformly  along  a  horizontal  plane :    What   is  the  co- 
efficient of  friction?  Ans.  £. 

2.  The  value  of  //,  is  .3,  the  weight  of  the  body  is  16  kilos : 
~What  force  is  required  to  move  it  uniformly?          -4ns.  4.8  kilos. 

3.  It  is  found  that  a  force  of  40  grams  suffices  to  move  a  body 
uniformly  on  a  horizontal  surface,  where  the  value  of  the  coefficient 
•of  friction  is  known  to  be  .25  :  What  is  the  weight  of  the  body  ? 

-4ns.  160  grams. 

4.  A  body  weighing  15  kilos  is  just  on  the  point  of  sliding 
-when  the  surface  it  rests  upon  is  inclined  20°  :  (a)  What  is  the  co- 
«fficient  of  friction  and  the  force  of  friction  ?  (b)  If  the  weight  of 
the  body  is  doubled,  what  values  have  these  quantities? 

j  (a)  /*  =  .36,  F  =  5.07  kilos. 
Ans'  ((&)/*  =  .36,  F=  10.14  kilos. 


CHAPTER    VII. 

MOTION  ON  INCLINED  PLANES.— THE  PENDULUM. 

150.  The  Force  which  Moves  a  Body  Down  an  In- 
clined Plane. — It  was  shown  (Art.  125)  that  when  the  force 
acts  in  a  line  parallel  to  the  inclined  plane,  P  :  W  ::  A  B  :  A  C. 
If,  therefore,  P  ceases  to  act,  the  body  descends  the  plane  only  with 
&  force  equal  to  P. 

Let  g  (the  velocity  acquired  in  a  second  in  falling  freely)  =  the 
force  of  gravity,  /  =  the  force  acting  down  the  plane,  h  =  the 
lieight,  I  =  the  length  ;  then  by  substitution, 
/:  g  ::  h  :  I,  and 

/=?-, 


FORMULAS    FOR    THE    INCLINED    PLANE.         97 

Therefore,  the  force  which  moves  a  body  down  an  inclined 
plane  is  equal  to  that  fraction  of  gravity  which  is  expressed  by 
the  height  divided  by  the  length.  This  is  evidently  a  constant 
force  on  any  given  plane,  and  produces  uniformly  accelerated 
motion.  Therefore  the  motion  on  an  inclined  plane  does  not 
differ  from  that  of  free  fall  in  kind,  but  only  in  degree.  Hence 
the  formulae  for  time,  space,  and  velocity  on  an  inclined  plane  are 
like  those  relating  to  free  fall,  if  the  value  of  /  be  substituted 
fot'ff, 

151.  Formulae  for  the  Inclined  Plane.— The  formulae 
for  free  fall  (Art.  27)  are  here  repeated,  and  against  them  the 
corresponding  formulae  for  descent  on  an  inclined  plane. 

Free  fall.  Descent  on  an  inclined  plane. 


2.      t  =  \f  11 

i 

s-  «=£ 


j     . 


6.      v  =  g  t 


By  formula  1,  s  <x  t2,  and  by  formula  3,  s  oo  v2.  It  follows  that 
in  equal  successive  times  the  spaces  of  descent  are  as  the  odd 
numbers,  1,  3,  5,  &c.,  and  of  ascent  as  these  numbers  inverted ; 
also,  that  with  the  acquired  velocity  continued  uniformly,  a  body 
moves  twice  as  far  as  it  must  descend  to  acquire  that  velocity.  If 
.a  body  be  projected  up  an  inclined  plane,  it  will  ascend  as  far  as  it 
must  descend  in  order  to  acquire  the  velocity  of  projection.  The 
distance  passed  over  in  the  time  t  by  a  body  projected  with  the 

velocity  v,  down  or  up  an  inclined  plane,  equals  t  v  ±  . 

152.  Formulae  for  the  whole  Length  of  a  Plane. — 

1.  The  velocity  acquired  in  descending  a  plane  is  the  same  as 
that  acquired  in  falling  down  its  height. 

For  now  s  =  I ;  hence  (formula  4),  v  =(^-j — -)  =  (  %  9  h  )  > 

which  is  the  formula  for  free  fall  through  h,  the  height  of  the 
plane. 

7 


MECHANICS. 


On  different  planes,  therefore,  v  oc  IVs. 

2.  The  time  of  descending  a  plane  is  to  the  time  of  falling  down 
its  height  as  the  length  to  the  height. 

For  (formula  2)  *  =          )  =  ?-.      But  the  time  of  fall 


down  the  height  is  (  —  j.     Therefore, 

t  down  plane  :  t  down  height  :  :  I  (—7)  :  (  —  -)   ', 


On  different  planes,  t  oc  — — :. 
Vh 

It  follows  that  if  several  planes  have  the  same  height,  the  ve- 
locities acquired  in  descending  them  are  equal,  and  the  times  of 
descent  are  as  the  lengths  of  the  planes.  For,  let  A  C,  A  D,  A  E^ 

(Fig.  106)  have  the  same  height  A  B\  then,  since  v  cch*,  and  h 

is  the  same  for  all.  v  is  the  same.     And  since  t  <x — -,  and  h  is 

Vh 
the  same  for  all  the  planes,  t  x  1.  FlG 


FIG.  106. 


153.  Descent  on  the  Chords  of  a  Circle.— In  descending 
the  chords  of  a  circle  which  terminate  at  the  ends  of  the  vertical 
diameter,  the  acquired  velocities  are  as  the  lengths,  and  the  times 
of  descent  are  equal  to  each  other  and  to  the  time  of  falling  through 
the  diameter. 

For  (Art.   152)   the  velocity  acquired  on  A  C  (Fig.  107)  = 

(»  ,  -  A,)*  =  (»,  4f)  *=  A  O(^f,  which,  since 
is  constant,  varies  as  A  C,  the  length. 


NO    LOSS    ON    A    CURVE.  99 

/2  A  C2^ 

Again   (Art.    152),  the    time    down    A    O  =  ^ — - )    = 

/2  A  B  .  A  c\l      /2A  8\l 

— )    ==( I  ,  which  is  equal  to  the  time  of  falling 

freely  through  A  B,  the  diameter. 

154.  Velocity  Acquired  on  a  Series  of  Planes. — If  no 

velocity  be  lost  in  passing  from  one  plane  to  another,  the  velocity 
acquired  in  descending  a  series  of  planes  is  equal  to  that  acquired 
in  falling  through  their  perpendicu-  FIG 

lar  height.  For,  in  Fig.  108,  the 
velocity  at  B  is  the  same,  whether 
the  body  comes  down  A  B  or  E  B, 
as  they  are  of  the  same  height,  FT). 
If,  therefore,  the  body  enters  on  B  C 
with  the  acquired  velocity,  then  it  is 
immaterial  whether  the  descent  is 
on  A  E  and  B  C  or  on  E  C ;  in  D 
either  case,  the  velocity  at  C  is  equal  to  that  acquired  in  falling  Fc. 
In  like  manner,  if  the  body  can  change  from  B  C  to  CD  without 
loss  of  velocity,  then  the  velocity  at  D  is  the  same,  whether  ac- 
quired on  A  B,  B  (7,  and  C  D,  or  on  F  D,  which  is  the  same  as 
down  F  G. 

165.  The  Loss  in  Passing  from  one  Plane  to  Another. 

— The  condition  named  in  the  foregoing  article  is  not  fulfilled.  A 
body  does  lose  velocity  in  passing  from  one  plane  to  another.  And 
the  loss  is  to  the  whole  previous  velocity  as  the  versed  sine  of  the 
angle  between  the  planes  to  radius. 

Let  B  F  (Fig.  109)  represent  the  velocity  which  the  body  has 
at  B.  Resolve  it  into  B  D  on  the  second  plane,  and  D  jP  perpen- 
dicular to  it.  B  D  is  the  initial  velocity  on  B  (7; 
and,  if  B  I  =  B  F,  D  I  is  the  loss.  But  D  I  is 
the  versed  sine  of  the  angle  F  B  D,  to  the  radius 
B  F;  and  /.  the  loss  is  to  the  velocity  at  B  as  D  I 
:  B  F : :  ver.  sin  B. :  rad. 

156.  No  Loss  on  a  Curve. — Suppose  now 
the  number  of  planes  in  a  system  to  be  infinite  ; 
then  it  becomes  a  curve  (Fig.  110).  As  the  angle 
between  two  successive  elements  of  the  curve  is  in- 
finitely small,  its  chord  is  also  infinitely  small ;  but 
its  versed  sine  is  infinitely  smaller  still,  i.  e.,  an  infi- 
nitesimal of  the  second  order  ;  for  diam.  :  chord  : :  chord  :  ver. 
sin.  Therefore,  although  the  sum  of  all  the  infinitely  small  angles 


100 


MECHANICS. 


is  a  finite  angle  180°  —  A  G  D,  yet,  as  the  loss  of  velocity  at  each 
point  is  an  infinitesimal  of  the  second  order,         FIG.  110.  A 

the  entire  loss  (which  is  the  sum  of  the  losses 
at  all  points  of  the  curve)  is  an  infinitesimal 
of  the  first  order. 

Hence,  a  body  loses  no  velocity  on  a  curve, 
and  therefore  acquires  at  the  bottom  the 
same  velocity  as  in  falling  freely  through  its 
height.  D  & 

It  appears,  therefore,  that  whether  a  body  descends  vertically, 
or  on  an  inclined  plane,  or  on  a  curve  of  any  kind,  the  acquired 
velocity  is  the  same,  if  the  height  is  the  same. 

157.  Times  of  Descending  Similar  Systems  of  Planes 
and  Similar  Curves. — If  planes  are  equally  inclined  to  the  hori- 
.  zon,  the  times  of  describing  them  are  as  the  square  roots  of  their 
lengths.    For,  if  the  height  and  base  of  each  plane  be  drawn,  simi- 
lar triangles  are  formed,  and  h  :  Zis  a  constant  ratio  for  the  several 

planes.     By  Art.  152,  t  oc  — —  oc  — —  oc  -v/T;  that  is,  the  time 
Vh          VI 

varies  as  the  square  root  of  the  length. 

If  two  systems  of  planes  are  similar,  i.  e.,  if  the  correspond- 
ing parts  are  proportional  and  equally  inclined  to  the  horizon,  it 
is  still  true  that  the  times  of  descending  them  are  as  the  square 
roots  of  their  lengths. 

abed  (Fig.  Ill)  be  similar,  and  let  A  F  and 


e    f 


a/be  drawn  horizontally,  and  the  lower  planes  produced  to  meet 
them,  then  it  is  readily  proved  that  all  the  homologous  lines  of 
the  figures  are  proportional,  and  their  square  roots  also  propor- 
tional. Then  (reading  t,  A  B,  time  down  A  B,  &c.), 


we  have 


t,  A  B:t,  a  b  :: 

t,  E-B  :  t,  e  b  :  : 


:  :  \/Zl?  :  Val  ; 


THE    PENDULUM.        ,  101 

and  t,EC'.t,ec•.\^/'WC\^/'ec\•.^fTB\^/Tb'1 

.-.  (by  subtraction)   t,  B  C:t,bc::  VAB  :  Vab. 
In  like  manner,  t,  CD  :  t,  cd  :  :  \/A  B  :  Va>  b. 
,-.  (by  addition) 

t>(A£  +  £C+  CD]  :t,  (a  I  +  be  +  cd)  ::VTB  :  Vab 

B  +  B  C  +  CD]  :  ^/(a  I  +  b  c  +  c  d). 


Though  there  is  a  loss  of  velocity  in  passing  from  one  plane 
to  another,  the  proposition  is  still  true  ;  because,  the  angles  being 
equal,  the  losses  are  proportional  to  the  acquired  velocities  ;  and 
therefore  the  initial  velocities  on  the  next  planes  are  still  in  the 
same  ratio  as  before  the  losses  ;  hence  the  ratio  of  times  is  not 
changed. 

The  reasoning  is  applicable  when  the  number  of  planes  in  each 
system  is  infinitely  increased,  so  that  they  become  curves,  similar, 
and  similarly  inclined  to  the  horizon.  Suppose  these  curves  to  be 
circular  arcs  ;  then,  as  they  are  similar,  they  are  proportional  to 
their  radii.  Hence,  the  times  of  descending  similar  circular  arcs 
are  as  the  square  roots  of  the  radii  of  those  arcs. 

158.  Questions  on  the  Motions  of  Bodies  on  Inclined 
Planes.  — 

1.  How  long  will  it  take  a  body  to  descend  100  feet  on  a  plane 
whose  length  is  150  feet,  and  whose  height  is  60  feet  ? 

Ans.  3.9  sec. 

2.  There  is  an  inclined  railroad  track,  2£  miles  long,  whose 
inclination  is  1  in  35.     What  velocity  will  a  car  acquire,  in  run- 
ning the  whole  length  of  the  road  by  its  own  weight  ? 

Ans.  106.2  miles  per  hour. 

3.  A  body  weighing  5  Ibs.  descends  vertically,  and  draws  a 
weight  of  6  Ibs.  up  a  plane  whose  inclination  is  45°.     How  far  will 
the  first  body  descend  in  10  seconds?  Ans.  110.74  feet. 

159.  The  Pendulum.  —  A  pendulum  is  a  weight  attached 
by  an  inflexible  rod  to  a  horizontal  axis  of  suspension,  so  as  to  be 
free  to  vibrate  by  the  force  of  gravity.     If  it  is  drawn  aside  from 
its  position  of  rest,  it  descends,  and  by  the  momentum  acquired, 
rises  on  the  opposite  side  to  the  same  height,  when  gravity  again 
causes  its  descent  as  before.     If  unobstructed,  its  vibrations  would 
never  cease. 

A  single  vibration  is  the  motion  from  the  highest  point  on  one 
side  to  the  highest  point  on  the  other  side.  The  motion  from 
the  highest  point  on  one  side  to  the  same  point  again  is  called  a 
•double  vibration. 


102 


MECHANICS. 


-I-  C 


The  axis  of  the  pendulum  is  a  line  drawn  through  its  centre  of 
gravity  perpendicular  to  the  horizontal  axis  about  which  the  pen- 
dulum vibrates. 

The  centre  of  oscillation  of  a  pendulum  is  that  point  of  its  axis 
at  which,  if  the  entire  mass  were  collected,  its  time  of  vibration 
would  be  unchanged. 

The  length  of  a  pendulum  is  that  part  of  its  axis  which  i& 
included  between  the  axis  of  suspension  and  the  centre  of  oscil- 
lation. 

All  the  particles  of  a  pendulum  may  be  conceived  to  be  col- 
lected in  points  lying  in  the  axis.  Those  which  are  above  the  cen- 
tre of  oscillation  tend  to  vibrate  quicker  (Art.  157),  and  therefore 
accelerate  it ;  those  which  are  below  tend  to  vibrate  slower,  and 
therefore  retard  it.  But,  according  to  the  definition  of  the  centre 
of  oscillation,  these  accelerations  and  retardations  exactly  balance 
each  other  at  that  point. 

160.  Calculation  of  the  Length  of  a  Pendulum,— Let 

G  q  (Fig.  112)  be  the  axis  of  a  pen- 
dulum in  which  all  its  weight  is  col-  FlG-  113-  Fia-  H& 
lected,  C  the  point  of  suspension, 
Cf  the  centre  of  gravity,  0  the  cen- 
tre of  oscillation,  a  b,  &c.,  particles 
above  0,  which  accelerate  it,  p,  q, 
&c.,  particles  below  0,  which  retard 
it.  GO  —  I,  is  the  length  of  the 
pendulum  required.  Denote  the 
masses  concentrated  in  a,  b  .  .  .  •  p,g, 
by  m,  m'  .  .  .  .  »i",  m'",  and  their 
distances  from  C  by  r,  r'  .  .  ,  .  r", 
r'"  ;  and  denote  the  distance  from 
C  to  G  by  k.  Denote  the  angular 
velocity,  that  is,  the  velocity  at 
unit's  distance  from  the  centre,  at 
any  instant  by  6  •  then  the  velocity 
of  m  will  be  r  6  and  its  momentum 
will  be  m  r  6. 

If  m  had  been  placed  at  0,  the 
momentum  would  have  been  ml  B. 
is  that  portion  of  the  force  which  accelerates  the  motion  of  the 
system. 

For  suppose  a  material  particle  m  (Fig.  113)  to  act  upon  a 
pendulum  C  D  without  weight ;  m  at  a  would,  under  the  action 
of  the  component  of  gravity  a  b,  move  the  point  a  to  b  and  swing 
the  pendulum  through  the  angle  Q  ;  m  if  transferred  to  o  would, 


-  -  c 

-  -o 

--P 


•Hr 

The  difference  m  (I  —  r)  6, 


THE    PENDULUM.  103 

gravity  being  the  same,  move  the  point  o  to  x,  and  swing  the 
pendulum  through  an  angle  less  than  8.  Thus  m  at  a  swings  the 
pendulum  through  a  greater  angle  in  a  given  time  than  it  would 
if  at  o,  or  accelerates  the  pendulum,  by  a  force  which  would  carry 
m  over  x  y  in  the  given  time,  or  by  m  (I  —  r)  0  ;  for,  calling 
C  a  =  r  and  C  o  =  I,  ab  =  r  6,  ox  =  ab=^r6,  o  y  =  I  0  ;  then 
o  y  —  ox  =  xy  =  ld  —  r  6  =  (I  —  r)  0,  and  m  moving  with 
velocity  x  y,  or  (I  —  r}  6,  gives  momentum  m  (I  —  r)  6. 

The  moment  of  this  force  with  respect  to  C  is  m  (I  —  r)  r  6. 

In  like  manner  the  moment  of  m'  is  m'  (I  —  r')  r'  6,  and  so  on 
for  all  the  particles  between  C  and  0. 

The  moments  of  the  forces  tending  to  retard  the  system 
applied  at  the  points  p,  q,  &c.,  are 

m"  (r"  —  I)  r"  d,  m'"  (r"'  —  1}  r"  d,  &o. 
But  since  these  forces  are  to  balance  each  other,  we  haye- 
m  (I  —  r)  r  6  +  m'  (I  —  r')  r'  6  +  &c.  =  m"  (r"  —  1)  r"  & 

+  m'"  (r'"  —  1)  r'"  8  +  &c. ; 

m  r2  +  m'  r'2  +  m"  r"2  +  &c. 

whence  I  =  —         — -. — r -.-. — T. 5 — 

m  r  +  m  r    +  m    r    +  &c. 

•Or  I  =  -oi—Hrj  where  S  denotes  the  sum  of  all  the  terms  similar 
S  (m  r) 

to  that  which  follows  it. 

The  numerator  of  this  expression  is  called  the  moment  of  inertia 
•of  the  body  with  respect  to  the  axis  of  suspension,  and  the  de- 
nominator is  called  the  moment  of  the  mass,  with  cespect  to  the 
.axis  of  suspension. 

By  the  principle  of  moments  m  r  +  m'  r'  +  &c.,  or  S  (m  r)  = 
M  k,  where  M  denotes  the  entire  mass  of  the  pendulum  ;  hence, 

by  substitution,  I  =  — -    .     • 

JM.  K 

That  is,  the  distance  from  the  axis  of  suspension  to  the  centre 
of  oscillation  is  found  by  dividing  the  moment  of  inertia,  with 
respect  to  that  axis,  by  the  moment  of  the  mass  with  respect  to  the 
same  axis. 

161.  The  Point  of  Suspension  and  the  Centre  of 
Oscillation  Interchangeable. — Let  the  pendulum  now  be  sus- 
pended from  an  axis  passing  through  0,  and  denote  by  I'  the 
•distance  from  0  to  the  new  centre  of  oscillation.  The  distances  of 
a,  b  .  .  .  .  p,  q.  from  0,  will  be  I  —  r,  I  —  r',  &c.,  and  the  distance 
O  0  will  be  I  —  k. 


104  MECHANICS. 

Hence,  from  the  principle  just  established,  we  have 
_  S  [m  (I  -  r)»]  _ 


But    from    the    preceding   paragraph    Z  =  ~  V^T  »  whence, 

S(mr*)  =  Mkl  ;  and  since  I  is  constant,  S(mP)  =  S(m  +  m'  -\~ 
m"  +  &c.)  f  =  Jf  T,  which  values  substituted  above  give 

,  _  MT  -2lS(mr)  +  Mkl  _  MF  -2  Mkl  +  Mkl 

M(l~-k)  ~~M(l  -  k) 

_M(l-k)l  = 
M(l-k) 

This  last  equation  shows  that  the  centre  of  oscillation  and  the 
point  of  suspension  are  interchangeable  ;  that  is,  if  the  pendulum 
were  suspended  from  0,  it  would  vibrate  in  the  same  time  as  when 
suspended  from  C. 

This  fact  is  taken  advantage  of  in  determining  the  length  of 
the  second's  pendulum  at  any  place.  A  solid  bar 
(Fig.  114)  is  furnished  with  two  knife-edge  axes  A 
and  B,  and  two  sliding  weights  G  and  D.  By  adjust- 
ing these  weights  the  bar  can  be  made  to  oscillate  in 
the  same  time  when  suspended  upon  either  axis. 
The  distance  between  the  knife-edges  A  and  B  is  the 
length  of  an  equivalent  simple  pendulum,  and  by 
comparing  .the  time  of  oscillation  with  that  of  a  pen- 
dulum beating  seconds,  the  time  of  one  oscillation 
of  this  reversible  pendulum  is  obtained  ;  from  these 
data  the  length  of  the  second's  pendulum  is  readily 
computed. 

162.  Calculation  of  the  Time  of  Oscilla- 
tion. —  Let  the  length  of  the  pendulum  AB  (Fig. 
115)  be  represented  by  I,  and  the  height  of  the  arc 
of  oscillation  by  B  D.  Suppose  the  pendulum  to 
have  moved  from  C  to  a  ;  its  acquired  velocity  will 
be  v  =  V  2g  x  D  H.  (Art.  156.) 

During  the  succeeding  infinitely  small  interval  of 
time  t'  it  will  describe  the  element  of  its  arc  a  c  with  the  velocity 
v  ;  hence 


V  20  x  DH 
Describe  upon  D  B  a,  semi-circumference ;  m  o  is  the  elemen- 


THE    PENDULUM. 


105 


FIG.  115. 


tary  arc  of  the  semi-circumference  having  the  same  altitude  E  H 
as  the  element  a  c.  As  these  arcs  are  elements  of  the  curves, 
they  may  be  regarded  as  straight 
lines,  and  a  b  c  and  m  n  o  become  tri- 
angles. A  H  a  and  a  b  c  are  similar 
triangles,  their  corresponding  sides 
being  perpendicular  two  and  two, 

a  c      A  a 

and  give  the  equation  —r  =  — JT> 

and  because  the  triangles  m  n  o  and 
Im  H  are   similar,  for  the  reason 

mo        I  m 

assigned  above, =  — =y  • 

'no       mH 

Divide   the  first  of  these  equa- 
tions  by  the   second,   and  we  get 

ac  A  a  x  m  H 

mo       a  H  x  Im 

But  all2  =  B  H  (2  A  B  -  B  H)  =  B  H  (2  I  —  B  H)  and 
2  =  B  H  x  H  D,  whence, 


ac_ 

mo 


x  HP 


,  or  since 


I  m  =  £  B  D  and  B  H  =  B  D  —  D  H, 


—  (BD  —  DH} 

and  this  substituted  in  the  value  of  t',  gives 


—  x  mo, 


—  (BD  —  DH} 
2  I  x  m  o 


^2g[2l  —  (BD  —  D  H)} 
or  dividing  both  terms  by  2  V  I,  we  obtain 
A/  I  x  mo 


it 
t  = 


BD  x  Vg  \/l- 


BD  —DH 

21 

When  the  amplitude  is  small,  the  double  arc  B  C  not  being  orer 

f>     T\  T)     TT 

5  degrees,   -^-y-,  and  are  very  small,  and  their  difference, 

&   i  '*    I 

— —j ,  is  smaller  still,  and  may  be  neglected,  giving  as  a 


106  MECHANICS. 

result  t'  =  \/  —  x  -77-77-  The  time  required  to  describe  C  B 
is  the  sum  of  the  times  of  describing  the  elements  of  C  B,  or 
calling  this  time  ^,  we  have  ^  =  \l  —  x  -^-jj  X  (sum  of  the 


elementary  arcs  m  o). 

But  the  sum  of  the  elements  of  D  m  B  corresponding  to  the 

7?  T} 

elements  of  C  B  is  the  semicircle  D  in  B  itself,  or  TT  —  —  ;  whence 


-  =  -  \/  —  =  time  of  semi-oscillation,  or  calling  t  the  time  of 
lave 

-VI- 


a  complete  oscillation,  we  have 


t 

163.  Applications  of  the  Formula. — From  the  equation 
t  =  TT  A  /  — ,  we  get  I  =  9—-'  Therefore,  the  length  of  a  pen- 
dulum being  known,  the  time  of  one  vibration  is  found  ;  and  on 
the  other  hand,  if  the  time  of  a  vibration  is  known,  the  length 
of  the  pendulum  is  obtained  from  it. 

From  the  same  formula,  we  find  that  t  oc  VI,  or 

The  time  in  which  a  pendulum  makes  a  vibration  varies  as  the 
square  root  of  the  length. 

As  t  cc  \/I,  .\  I  oc  tz ;  hence,  if  the  length  of  a  seconds  pendu- 
lum equals  I,  then  a  pendulum  which  vibrates  once  in  tivo  seconds 
equals  4  I,  and  one  which  beats  half  seconds  •=  \l,  &c. 

Again,  by  observing  the  length  of  a  pendulum  which  vibrates 
in  a  given  time,  the  force  of  gravity,  g,  may  be  found.  For,  as 

/  =  ~j-,  /.  g  =  -TJ- '     And  if  g  varies,  as  it  does  in  different  lati- 
tudes and  at  different  altitudes,  then  Z  =  ^  QC  g  t3 ;   and  if  the 

time  is  constant  (as,  for  example,  one  second),  then  I  oc  g.    Hence, 
The  length  of  a  pendulum  for  beating  seconds  varies  as  the  force 
of  gravity. 

Also,  t  oc  (  — }  ;  that  is,  the  time  of  a  vibration  varies  di- 
rectly as  the  square  root  of  the  length,  and  inversely  as  the  square 
root  of  the  force  of  gravity. 

Since  the  number,  n,  of  vibrations  in  a  given  time  varies 

inversely  as  the  time  of  one  vibration,  therefore  n  oc  (~\  ,   and 


THE    PENDULUM. 


107 


g  oc  In2.     Hence,  if  the  time  and  the  length  of  a  pendulum  are 
given, 

The  force  of  gravity  varies  as  the  square  of  the  number  of  vibra- 
tions. 

1.  What  is  the  length  of  a  pendulum  to  beat  seconds,  at  the 
place  where  a  body  falls  16^  ft.  in  the  first  second? 

Ans.  39.11  inches,  nearly. 

2.  If  39.11  inches  is  taken  as  the  length  of  the  seconds  pendu- 
lum, how  long  must  a  pendulum  be  to  beat  10  times  in  a  minute? 

Ans.  m^feet. 

3.  In  London,  the  length  of  a  seconds  pendulum  is  39.1386 
inches ;  what  velocity  is  acquired  by  a  body  falling  one  second  in 
that  place  ?  Ans.  32.19  feet. 

164.  The  Compensation  Pendulum. — This  name  is  given 
to  a  pendulum  which  is  so  constructed  that  its  length  does  not 
vary  by  changes  of  temperature.  As  all  substances  expand  by 
heat  and  contract  by  cold,  therefore  a  pendulum  will  vibrate  more 
slowly  in  warm  than  in  cold  weather.  This  difficulty  is  overcome 
in  several  ways,  but  always  by  employing  two 
substances  whose  rates  of  expansion  and  con- 
traction are  unequal.  One  of  the  most  com- 
mon is  the  gridiron  pendulum,  represented  in 
Fig.  116.  It  consists  of  alternate  rods  of 
steel  and  brass,  connected  by  cross-pieces  at 
top  and  bottom.  The  rate  of  longitudinal 
expansion  and  contraction  of  brass  to  that 
of  steel  is  about  as  100  to  61 ;  so  that  two 
lengths  of  brass  will  increase  and  diminish 
more  than  three  equal  lengths  of  steel.  There- 
fore, while  there  are  three  expansions  of  steel 
downward,  two  upward  expansions  of  brass 
can  be  made  to  neutralize  them.  In  the 
figure  the  dark  rods  represent  steel,  the  white 
ones  brass.  Suppose  the  temperature  to  rise, 
the  two  outer  steel  rods  (acting  as  one)  let 
down  the  cross-bar  d ;  the  two  brass  rods 
standing  on  d  raise  the  bar  b ;  the  steel  rods 
suspended  from  b  let  down  the  bar  e,  on 
which  the  inner  brass  rods  stand,  and  raise  the  short  bar  c ;  and 
finally,  the  centre  steel  rod,  passing  freely  through  d  and  e,  lets 
down  the  disk  of  the  pendulum.  These  lengths  (counting  each 
pair  as  a  single  rod)  are  adjusted  so  as  to  be  in  the  ratio  of  100 
for  the  steel  to  61  for  the  brass ;  in  which  case  the  upward 
expansions  just  equal  those  which  are  downward,  and  therefore 


108  MECHANICS. 

the  centre  of  oscillation  remains  at  the  same  distance  from  the 
point  of  suspension. 

If  the  temperature  falls,  the  two  contractions  of  brass  are  equal 
to  the  three  of  steel,  so  that  the  pendulum  is  not  shortened  by 
cold.  j> 

The  mercurial  pendulum  consists  of  a  steel  rod  terminating  at 
the  bottom  with  a  rectangular  frame  in  which  is  a  tall  narrow  jar 
containing  mercury,  which  is  the  weight  of  the  pendulum.  It 
requires  only  6.31  inches  of  mercury  to  neutralize  the  expansions 
and  contractions  of  42  inches  of  steel.  See  Appendix  for  calcula- 
tions of  the  place  of  the  centre  of  oscillation. 


CHAPTEE    VIII. 

CENTRAL    FORCES. 

165.  Central  Forces  Described. — When  two  forces  act  upon 
a  body — one  an  impulse,  which  alone  would  cause  uniform  motion 
in  a  straight  line ;  the  other  a  continued  force,  which  urges  the 
body  toward  some  point  not  in  the  line  of  motion  due  to  the  im- 
pulse— the  resultant  motion  will  be  in  a  curve.  The  first  is  called 
the  projectile  force,  the  second  the  centripetal  force. 

Suppose  a  point  m  (  Fig  117)  to  be  acted  upon  by  an  impulse, 

in  direction  and  intensity  re- 

,    ,  J  ,  FIG.  117. 

presented  by  o  m,  and  also  by 

a  constant  force,  m  d.  This 
centripetal  force  m  d  may  be 
resolved  into  two  components; 
one  m  a  in  the  direction  of 
the  tangent,  the  other,  m  h, 
perpendicular  to  it.  The  tan- 
gential component  will  accel- 
erate or  retard  the  motion  in  the  curved  path  according  as  it  acts 
with  the  projectile  force,  or  in  opposition  to  it,  while  the  com- 
ponent at  right  angles  to  this  tends  to  deflect  the  body  from  a 
rectilinear  path,  and  therefore  determines  the  character  of  the 
curve  at  any  instant. 

When  the  body  moves  in  the  circumference  of  a  circle,  the 
tangential  component  of  the  centripetal  force  is  0,  and  hence  the 
motion  is  uniform. 

If  the  centripetal  force  should  cease  to  act  at  any  instant,  the 


CENTRIFUGAL    FORCE.  101> 

body,  by  its  inertia,  would  immediately  begin  to  move  in  a  straight 
line  tangent  to  the  curve  at  the  point  where  the  body  was  when 
the  force  ceased  to  act. 

Since  the  body,  by  its  inertia,  tends  to  move  in  a  tangent,  there 
is  a  continued  resistance  to  deflection  into  a  curved  path,  equal 
and  opposed  to  the  component  m  h,  in  the  direction  of  the  radius 
of  curvature  at  the  instant ;  this  is  called  the  centrifugal  force,  and, 
like  gravity,  is  an  accelerating  force. 

166.  Expressions  for  the  Centrifugal  Force  in  Circu- 
lar Motion. — 

1.  Let  r  —  the  radius  of  the  circle,  v  =  FIG.  118. 

the  velocity  of  the  body,  /  =  the  accelera- 
tion of  the  centrifugal  force,  and  let  A  B 
(Fig.  118)  be  the  arc  described  in  the  in- 
finitely small  time  t  ;  then  A  B  —  v  t,  and 
by  a  method  similar  to  that  employed  in 
the  discussion  of  the  force  of  gravity,  it 
may  be  shown  that  A  E  =  B  D  -—  %f  t\ 

But  A  B,  being  a  very  small  arc,  may 
be  considered  as  equal  to  its  chord,  which 
is  a  mean  proportional  between  A  E  and  the  diameter  2  r.  Hence 


Reducing  we  have  the  acceleration  of  the  centrifugal  force 

/=£  ...........  <» 

From  (1)  it  follows,  that  in  equal  circles  the  acceleration  varies  as 
the  square  of  the  velocity. 

2.  The  value  of  /  may  be  expressed  in  a  different  form.  Let 
t'  =  the  time  of  a  complete  revolution,  then  2  TT  r  =  v  t'  ;  whence 

2  TT  r 
v  —  —  ;  —  •     This  substituted  in  (1)  gives 


Hence,  The  acceleration  varies  directly  as  the  radius  of  the  circler 
and  inversely  as  the  square  of  the  time  of  revolution. 

3.  If  the  acceleration  /is  imparted  to  a  mass  m,  the  centrifugal 
force  exerted  is  (Art.  12), 

_,  .      mv*    • 

F  =  mf=~- 

Introducing  the  weight  w  =  m  g,  we  have 

f  =  ^  ............     (3) 

g  r 


110 


MECHANICS. 


Let  n  —  the  number  of  revolutions  per  second  ;  then 
v  =  2  TT  r  n,  and  (3)  becomes 
F=  —.w.r.n* (4) 


9 


167.    Two    Bodies    Revolving   about  their   Centre   of 

Gravity. — Let  A  and  B  (Fig.  119)  be  two  masses  connected  by 

a  rod,  and  let  them  be  made  to 

revolve    about    the    centre   of  FIG-  119. 

gravity    C ;    then    by   (4)    the 

centrifugal  force  of  A  will  be 


— - .  A  .  A  G .  n\  and  of  B,  - 


B  .  B  C  .  n\ 


FIG.  120. 


But  C  being  the  centre  of  gravity  of  the  two  bodies,  A.AG  — 
B  .  B  C ;  .:  the  centrifugal  force  of  A  equals  that  of  B.  Hence, 
If  two  bodies  revolve  in  the  same  time  about  an  axis  passing  through 
their  centre  of  gravity,  there  will  be  no  strain  upon  that  axis. 

168.  Centrifugal  Force  on  the  Earth's  Surface.— As  the 
«arth  revolves  upon  its  axis  all  free  particles  upon  it  are  influenced 
by  the  centrifugal  force.  Let  N  S  (Fig.  120)  be  the  axis,  and  A  a 
particle  describing  a  circumference  with  the  radius  A  0.  Put  r  = 
CQ,  r'  =  A  0, 1  =  the  angle  A  C  Q, 
the  latitude,  /  —  the  centrifugal  ac- 
celeration at  the  equator,  /'  =  the  -W 
acceleration  at  A,  v  —  velocity  of  Q, 
and  v'  =  velocity  of  A  ;  then 

f  —  W*       Af  —  V" 
Butu  :  v'  ::  r  :  r' ;  whence  vf  =  ' 
—     Again,  from  the  triangle  A  CO 

we  have  r'  —  r  cos  I ;  hence  v'  = 

.        ,  .,       v2  cos2  I       v1  cos  I 

v  cos  I,  and  f  = ,—r  =   , — • 

r  cos  I          .   r 

with  that  of/,  we  have 

/  =  /  cos  I. 

That  is,  the  centrifugal  force  at  any  point  on  the  earth's  surface  is 
equal  to  the  centrifugal  force  at  the  equator,  multiplied  by  the  cosine 
of  the  latitude  of  the  place. 

A  body  at  the  equator  loses  by  centrifugal  force  -^-g-  part  of 
the  weight  which  it  would  have  if  the  earth  did  not  revolve  on 
its  axis. 

Let  A  B  represent  the  centrifugal  force  at  A,  and  resolve  it 


Comparing  the  value  of  f 


CENTRAL    FORCES.  HI 

into  A  D  on  C  A  produced,  and  A  F,  tangent  to  the  meridian 
NQ  S;  then,  since  the  angle  DAB^AGQ  =  l,  we  have 

A  D  =  A  B  cos  I  =  /  cos  I .  cos  I  =  f  cos2  /. 

That  is,  that  component  of  the  centrifugal  force  at  any  point,  which 
opposes  the  force  of  gravity,  is  equal  to  the  centrifugal  force  at  the 
equator,  multiplied  by  the  square  of  the  cosine  of  the  latitude  of  the 
place. 

In  like  manner  we  find  A  F=  A  B  sin   I  =f  cos  I  sin  I  — 

- — ^ From  this  equation  we  see  that  the  tangential  com- 
ponent is  0  at  the  equator,  increases  till  I  =  45°  ;  where  it  is  a 
maximum  ;  then  goes  on  diminishing  till  I  —  90°,  when  it  again 
becomes  0. 

The  effect  of  A  D  is  to  diminish  the  weight  of  the  particle, 
while  the  effect  of  A  F  is  to  urge  it  toward  the  equator. 

169.  Examples  on  Central  Forces. — 

1.  A  ball  weighing  10  Ibs.  is  whirled  around  in  a  circumference 
of  10  feet  radius,  with  a  velocity  of  30  feet  per  second.     What  is 
the  tension  upon  the  cord  which  restrains  the  ball  ? 

Ans.  28  Ibs.,  nearly. 

2.  With  what  velocity  must  a  body  revolve  in  a  circumference 
of  5  feet  radius,  in  order  that  the  centrifugal  force  may  equal  the 
weight  of  the  body?  .  Ans.  v  =  12.7  ft. 

3.  A  ball  weighing  2  Ibs.  is  whirled  round  by  a  sling  3  feet 
long,  making  4  revolutions  per  second.     What  is  its  centrifugal 
force  ?  Ans.  117.84  Ibs. 

4.  A  weight  of  5  Ibs.  is  attached  to  the  end  of  a  cord   3  feet 
long  just  capable  of  sustaining  a  weight  of   100  Ibs.     How  many 
revolutions  per  second  must  the  body  make  in  order  that  the  cord 
may  be  upon  the  point  of  breaking?  Ans.  n  =  2.3,  nearly. 

5.  A  railway  carriage,  weighing   7  tons,  moving  at  the  rate  of 
30  miles  per  hour,  describes  an  arc  whose  radius  is  400  yards. 
What  is  the  outward  pressure  upon  the  track  ?  Ans.  701  +  Ibs. 

170.  Composition  of  two  Rotary  Motions. — 

When  a  body  is  rotating  on  an  axis,  and  a  force  is  applied  which 
alone  would  cause  it  to  rotate  on  some  other  axis,  the  body  will  com- 
mence rotation  on  an  axis  lying  betioeen  them,  and  the  velocities  of 
rotation  on  the  three  axes  are  such,  that  each  may  be  represented  by 
the  sine  of  the  angle  between  the  other  two. 

Suppose  a  body  is  rotating  on  an  axis  A  B  in  the  plane  of  H  K, 
and  that  a  force  is  applied  to  make  it  rotate  on  the  axes  G  D  in  the 
same  plane  H  K,  these  two  axes  intersecting  within  the  body  at 
some  point  called  G. 


MECHANICS. 


FIG.  121. 


Imagine  a  perpendicular  to  the  plane  of  the  axes  to  be  drawn 
through  G,  and '  let  P  be  a  particle  of 
the  body  in  this  perpendicular.  Sup- 
pose the  particle  P,  in  an  infinitely 
small  time  t,  to  pass  over  P  a  perpen- 
dicular to  A  B,  by  the  first  rotation, 
and  over  P  c,  perpendicular  to  C  D,  by 
the  second.  Then,  since  the  particle 
wfll  describe  the  diagonal  P  e  in  the 
time  t,  this  line  must  indicate  the 
direction  and  velocity  of  the  resultant 
rotation.  Therefore,  if  E  F  be  drawn 
through  G,  perpendicular  to  the  plane 
G  P  e,  E  F  is  the  axis  on  which  the 
body  revolves  in  consequence  of  the  two  rotations  given  to  it. 
Since  P  G  is  perpendicular  to  the  plane  A  G  C,  and  also  to  the 
line  E  F,  therefore  E  Fis  in  that  plane  ;  that  is,  the  new  axis  of 
rotation  is  in  the  plane  of  the  other  two  axes.  The  angles  AGE 
and  E  G  C,  are  respectively  equal  to  the  angles  a  P  e  and  e  P  c, 
the  inclinations  of  the  planes  of  rotation.  But  the  lines,  P  a, 
PC,  P  e,  represent  the  velocities  in  those  directions  respectively ; 
and  (Art.  44)  P  a  :  P  c  :  P  e  : :  sin  c  P  e  :  sin  a  P  e:  sin  a  P  c  ; 
therefore  P  a  :  P  c  :  P  e  : :  sin  C  G  E  :  sin  A  G  E  :  sin  A  G  C; 
or,  the  velocities  on  the  three  axes,  (namely,  the  axes  of  the  com- 
ponent rotations,  and  of 
the  resultant  rotation,) 
.  are  such,  that  each  may 
be  represented  by  the 
sine  of  the  angle  between 
the  other  two  axes. 

171.  The  Gyro- 
scope. — The   gyroscope 

"*;**  affords    an    illustration 

!     of    the   composition    of 
y      two  rotations  imparted 
to  a  body.     As   usually 
X'  constructed,   it   consists 

of  a  heavy  wheel  G  H 
(Fig.     122),    accurately 
balanced   on     the     axis 
a  b,    which    runs  with 
as  little  friction  as  pos- 
sible upon  pivots  in  a  metallic  ring.     In  the  direction  of  the 
axis,  there  is  a  projection  B  from  the  ring,  having  a  socket 


THE    GYROSCOPE.  113 

sunk  into  it  on  the  under  side,  so  that  it  may  rest  on  the  pointed 
standard  S,  without  danger  of  slipping  off. 

The  wheel  is  made  to  rotate  swiftly  by  drawing  off  a  cord 
ivound  upon  a  b,  and  then  the  socket  in  B  is  placed  on  the 
standard,  and  the  whole  left  to  itself.  Immediately,  instead  of 
falling,  the  ring  and  wheel  commence  a  slow  revolution  in  a 
horizontal  plane  around  the  standard,  the  point  A  following  the 
circumference  A  E  F,  in  a  direction  contrary  to  the  motion  of  the 
top  of  the  wheel. 

This  revolution  is  explained  by  applying  the  principle  of  com- 
position of  rotations  given  in  the  preceding  article.  The  particles 
of  the  wheel  are  rotating  about  the  horizontal  axis  a  J  by  the  force 
imparted  by  the  string.  The  force  of  gravity  tends  to  make  it 
fall,  that  is,  to  revolve  in  a  vertical  circle  around  the  axis  C  D  at 
right  angles  to  a  J.  Hence,  in  a  moment  after  dropping  the  ring, 
the  system  will  be  found  revolving  on  an  axis  which  lies  in  the 
direction  E  B,  between  A  B  and  Cf  D,  the  other  two  axes.  Now, 
gravity  bears  it  down  around  a  new  axis  perpendicular  to  E  B. 
Therefore,  as  before,  it  changes  to  still  another  axis  F  B,  and  thus 
continues  to  go  round  in  a  horizontal  circle. 

The  only  way  possible  for  it  to  rotate  on  an  axis  in  a  new  posi- 
tion, is  to  turn  its  present  axis  of  rotation  into  that  position. 
Hence,  the  whole  instrument  turns  about,  in  order  that  its  axis 
may  take  these  successive  positions. 

The  change  of  axis  is  seen  also  by  observing  the  resultant  of 
the  motions  of  the  particles  at  the  top  and  bottom  of  the  wheel. 
For  example,  G  is  moving  swiftly  in  the  direction  m  by  the  rota- 
tion around  a  ft ;  by  gravity  it  tends  to  move  slowly  in  the  line  r, 
tangent  to  a  vertical  circle  about  the  centre  B.  The  resultant  is 
in  the  line  n,  tangent  to  the  wheel  when  its  axis  a  b  has  taken  the 
new  position  E  B. 

The  centre  of  gravity  of  the  ring  and  wheel  tends  to  remain  at 
rest,  while  the  resultant  of  the  two  rotations  carries  around  it  all 
other  parts,  standard  included,  in  horizontal  circles.  But  the 
standard  by  its  inertia  and  friction  resists  this  effort,  and  the  re- 
action causes  the  ring  and  wheel  to  go  around  the  standard. 


PART  II. 

HYDRA.ULIC8. 


CHAPTEK   I. 

HYDROSTATICS. 

172.  Liquids  Distinguished  from  Solids  and  Gases. — 

A  fluid  is  a  substance  whose  particles  are  moved  among  each  other 
by  a  very  slight  force.  In  solid  bodies  the  particles  are  held  by 
the  force  of  cohesion  in  fixed  relations  to  each  other;  hence  such 
bodies  retain  their  form  in  spite  of  gravity  or  other  small  forces 
exerted  upon  them.  If  a  solid  be  reduced  to  the  finest  powder, 
still  each  grain  of  the  powder  is  a  solid  body,  and  its  atoms  are 
held  together  in  a  determinate  shape.  A  pulverized  solid,  if  piled 
up,  will  settle  by  the  force  of  gravity  to  a  certain  inclination,  ac- 
cording to  the  smallness  and  smoothness  of  its  particles,  while  a 
liquid  will  not  rest  till  its  surface  is  horizontal. 

Fluids  are  of  two  kinds,  liquids  and  gases.  ID  a  liquid,  there 
is  a  perceptible  cohesion  among  its  particles  ;  but  in  a  gas,  the 
particles  mutually  repel  each  other.  These  fluids  are  also  distin- 
guished by  the  fact  that  liquids  cannot  be  compressed  except  in  a 
very  slight  degree,  while  the  gases  are  very  compressible.  A  force 
of  15  pounds  on  a  square  inch,  applied  to  a  mass  of  water,  will 
compress  it  only  about  .000046  of  its  volume,  as  is  shown  by  an 
instrument  devised  by  Oersted.  But  the  same  force  applied  to  a 
quantity  of  air  of  the  usual  density  at  the  earth's  surface  will  re- 
duce it  to  one-half  of  its  former  volume. 

173.  Transmitted  Pressure.— It  is  an  observed  property  of 
fluids  that  a  force  which  is  applied  to  one 

T^TYX       1 9^i 

part  is  transmitted  undivided  to  all  parts. 
For  instance,  if  a  piston  A  (Fig.  123)  is 
pressed  upon  the  water  in  the  vessel  ADC 
with  a  force  of  one  pound,  every  other  pis- 
ton of  the  same  size,  as  B,  C,  Z>,  or  E,  re- 
ceives a  pressure  of  one  pound  in  addition 
to  the  previous  pressure  of  the  water  itself. 
Hence  the  whole  amount  of  bursting  pres- 
sure exerted  within  the  vessel  by  the  weight 


THE    HYDRAULIC    PRESS. 


115 


upon  A  equals  as  many  pounds  as  there  are  portions  of  surface 
equal  to  the  area  of  A.  And  if  the  pressure  is  increased  till  the 
vessel  bursts,  the  fracture  is  as  likely  to  occur  in  some  other  part 
as  in  that  toward  which  the  force  is  directed. 

174.  The  Hydraulic  Press. — An  important  application  of 
the  principle  of  transmitted  pressure  occurs  in  Bramah's  hydraulic 
press,  represented  in  Fig.  124.  The  walls  of  the  cylinder  and 

FIG.  124. 


reservoir  are  partly  removed,  to  show  the  interior.  A  is  a  small 
forcing  pump,  worked  by  the  lever  M,  by  which  water  is  raised  in 
the  pipe  a  from  the  reservoir  H,  and  driven  through  the  tube  K 
into  the  cylinder  B,  where  it  presses  up  the  piston  P,  and  the  iron 
plate  on  the  top  of  it,  against  the  substance  above.  Attach  down- 
ward stroke  of  the  small  piston  p,  a  quantity  of  water  is  trans- 
ferred to  the  cylinder  B,  and  presses  up  the  large  piston  with  a 
force  as  many  times  greater  than  that  exerted  on  the  small  one  as 
the  under  surface  of  P  is  greater  than  that  of  p  (Art  173).  II 
the  diameter  of  p  is  one  inch,  and  that  of  P  is  ten  inches,  then 
any  pressure  on  p  exerts  a  pressure  100  times  as  great  on  P>  The 
lever  M  gives  an  additional  advantage.  If  the  distances  from  the 
fulcrum  to  the  rod  p  and  to  the  hand  are  as  1  :  5,  this  ratio  com- 
pounded with  the  other,  1  :  100,  gives  the  ratio  of  power  at  M  to 


116  HYDROSTATICS. 

the  pressure  at  Q  as  1  :  500 ;  so  that  a  power  of  100  Ibs.  exerts  a 
pressure  of  50000  Ibs. 

This  machine  has  the  special  advantage  of  working  with  a 
small  amount  of  friction.  It  is  used  for  pressing  paper  and 
books,  packing  cotton,  hay,  &c. ;  also  for  testing  the  strength  of 
cables  and  steam-boilers.  It  has  been  sometimes  employed  to  raise 
great  weights,  as,  for  instance,  the  tubular  bridge  over  the  Menai 
straits ;  the  two  portions,  after  being  constructed  at  the  water 
level,  were  raised  more  than  100  feet  to  the  top  of  the  piers,  by 
two  hydraulic  presses.  The  weight  of  each  length  lifted  at  once 
was  more  than  1800  tons. 

The  relation  of  power  to  weight  in  the  hydraulic  press  is  in 
accordance  with  the  principle  of  virtual  velocities  (Art.  141). 
For,  while  a  given  quantity  of  water  is  transferred  from  the  smaller 
to  the  larger  cylinder,  the  velocity  of  the  large  piston  is  as  much 
less  than  that  of  the  small  one  as  its  area  is  greater.  But  we  have 
seen  that  the  pressures  are  directly  as  the  areas.  Therefore,  in 
this  as  in  other  machines,  the  intensities  of  the  forces  are  inversely 
as  their  virtual  velocities. 

Ex.  1.  A  press  of  the  same  form  as  in  Fig.  124  has  a  piston 
whose  cross  section  is  one  sq.  ft. ;  the  feed-pump  piston  is  2  sq.  in. 
cross  section,  and  stroke  6  inches.  The  lever  has  a  short  arm  of 
1  ft.  and  long  arm  of  4  ft.  (measured  from  fulcrum  in  each  case), 
Find  the  greatest  pressure  that  can  be  produced  by  a  man  who 
exerts  a  force  of  174  Ibs.,  friction  and  difference  of  level  of  the 
liquid  in  the  cylinders  being  disregarded.  Ans.  50112  Ibs. 

Ex.  2.  How  many  strokes  of  the  pump  will  it  take  to  raise  the 
press  piston  one  foot  in  the  last  example  ?  Ans.  144. 

175.  Equilibrium  of  a  Fluid. — In  order  that  a  fluid  may 
be  at  rest, 

1.  Tlie  pressures  at  any  one  point  must  be  equal  in  all  direc- 
tions. 

2.  The  surface  must  be  perpendicular  to  the  resultant  of  the 
forces  ivliicli  act  upon  it. 

Both  of  these  conditions  result  from  the  mobility  of  the  par- 
ticles.   It  is  obvious  that  the  first  must  be  true,  since,  if  any 
particle  were  pressed  more  in  one  direction  than  another,  it  would 
move  in  the  direction  of  the  greater 
force,  and  therefore  not  be  at  rest,  Fl°- 125- 

as  supposed. 

In  order  to  show  the  truth  of  the 
second  condition,  let  mp  (Fig.  125) 
represent  the  resultant  of  the  forces 
•which  act  on  the  fluid.  Then,  if  the 


CURVATURE    OF    A    LIQUID    SURFACE.  117 

surface  is  not  perpendicular  to  m  p,  that  force  may  be  resolved 
into  m  q  perpendicular  to  the  surface,  and  m  f  parallel  to  it. 
The  latter,  m  f,  not  being  opposed,  the  particles  move  in  that 
direction. 

As  gravity  is  the  principal  force  which  acts  on  all  the  par- 
ticles, the  surface  of  a  fluid  at  rest  is  ordinarily  level,  that  is,  per- 
pendicular to  a  vertical  or  plumb  line.  If  the  surface  is  of  small 
extent,  it  is  sensibly  a  plane,  though  it  is  really  curved,  because 
the  vertical  lines,  to  which  it  is  perpendicular,  converge  toward 
the  centre  of  the  earth. 

176.  The  Curvature  of  a  Liquid  Surface.  —  The  earth 
being  7912  miles  in  diameter,  a  distance  of  100  feet  on  its  surface 
subtends  an  angle  of  about  one  second  at  the  centre,  and  there- 
fore the  levels  of  two  places  100  feet  apart  are  inclined  one  second 
to  each  other. 

The  amount  of  depression  for  moderate  distances  is  found  by 
the  formula,  d  =  f  L*.  in  which  d  is  the  de- 
pression in  feet,  and  L  the  length  of  arc  in 
miles.  Let  B  E  (Fig.  126)  be  a  small  arc  of 
a  great  circle  on  the  earth  ;  then  C  E  is  the 
depression.  As  B  E  is  small,  its  chord  may 
be  considered  equal  to  the  arc,  and  B  G  equal 
to  the  depression.  But  B  G  :  B  E  :  :  B  E  : 

B  A  ;  that  is,d:L::L:  7912  ;  or  d  = 


In  order  to  express  d  in  feet,  while  the  other 
lines  are  in  miles,  we  have 

Z2  x  52802        L2  x  5280 

fj  — _ — " —  2    rs  vprv  npflrlv 

—  ™To  ^  ^')«a  —        7912         ~  f  ^  >  ^C1>  n  dny- 


This  gives,  for  one  mile,  d  =  8  inches  ;  for  two  miles,  d  =  2 
ft.  8  in.;  and  for  100  miles,  d  =  6667  ft,  &c.  If  a  canal  is  100 
miles  long,  each  end  is  more  than  a  mile  below  the  tangent  to  the 
surface  of  the  water  at  the  other  end. 

177.  The  Spirit  Level.  —  Since  the  surface  of  a  liquid  at 
rest  is  level,  any  straight  line  which  is  placed  parallel  to  such  a 
surface  is  also  level.  Leveling  instruments  are  constructed  on 
this  principle.  The  most  accurate  kind  is  the  one  called  the 
spirit  level.  Its  most  essential 
part  is  a  glass  tube,  A  B  (Fig.  Fro.  127. 

127),  nearly  filled  with  alcohol     B  <gH^_mzl__    ^>A 
(because  water  would  be  liable 

to  freeze),  and  hermetically  sealed.     The  tube  having  a  little  con- 
vexity upward  from  end  to  end,  though  so  slight  as  not  to  be 


118 


HYDROSTATICS. 


visible,  the  bubble  of  air  moves  to  the  highest  part,  -and  changes 
its  place  by  the  least  inclination  of  the  tube.  The  tube  is  so  con- 
nected with  a  straight  bar  of  wood  or  metal,  as  D  C  (Fig.  128),  or 
for  nicer  purposes,  with  FlG 

a  telescope,  that  the 
bubble  is  at  the  middle 
M  when  the  bar  or  the 
axis  of  the  telescope  is 
exactly  level.  The  tube 
usually  has  graduation  lines  upon  it  for  adjusting  the  bubble 
accurately  to  the  middle. 

178.  Pressure  as  Depth. — From  the  principle  of  equal 
transmission  of  force  in  a  fluid,  it  follows  that,  if  a  liquid  is  uni- 
formly dense,  its  pressure  on  a  given  area  varies  as  the  perpen- 
dicular depth,  whatever  the  form  or  size  of  the  reservoir.  Let  the 
vessel  A  B  C  D  (Fig.  129),  having  the  form  of  a  right  prism,  be 
filled  with  water,  and  imagine  the  water  to  be  divided  by  horizon- 
tal planes  into  strata  of  equal  thickness.  If  the  density  is  every- 
"where  the  same,  the  weights  of  these  strata  are  equal.  But  the 
pressure  on  each  stratum  is  the  sum  of  the  weights  of  all  the  strata 
above  it.  Therefore,  in  this  case,  the  pressure  varies  as  the  depth. 


FIG.  129. 


FIG.  130. 


A    B 


But  let  the  reservoir  (Fig.  130)  contain  water.  The  pressure 
of  the  column  A  B  L  Mia  transmitted  equally  in  every  direction 
(Art.  173).  If  the  area  of  the  section  L  M  is  one  sq.  inch  and 
the  weight  of  the  column  A  L  is  one  pound,  then  every  square 
inch  of  the  side  E  H  will  receive  a  pressure  of  one  pound,  on 
account  of  the  column  A  L,  in  addition  to  the  pressure  it  sus- 
tains from  the  contained  water.  So  also  every  square  inch  of  the 
bottom  D  H  will  sustain  an  added  pressure  of  1  lb.,  and  also  every 
square  inch  of  the  top  ME  will  sustain  an  upward  pressure 
of  1  lb.  That  is  to  say,  the  added  pressure  upon  every  part  of 
the  containing  vessel  L  E  H  D,  whose  area  equals  the  area  of  the 
base  of  the  column  A  L  M  B,  is  equal  to  the  weight  of  that 
column. 

Again,  if  the  base  is  smaller  than  the  top,  as  in  the  vessel 


PRESSURE    AS    DEPTH. 


119 


A  B  E  F  (Fig.  131),  then  the  pressure  on  E  F  equals  only  tfie 
weight  of  the  column  C  D  E  F.  The  water  in  the  surrounding 
space  A  C  E,  B  D  F,  simply  serves  as  a  vertical  wall  to  balance 
the  lateral  pressures  of  the  central  column. 

If  the  surface  pressed  upon  is  oblique  or  vertical,  then  the 
points  of  it  are  at  unequal  depths  ;  in  this  case,  the  depth  of  the 
area  is  understood  to  be  the  average  depth  of  all  its  parts ;  that  is, 
the  depth  of  its  centre  of  gravity. 

If  the  fluid  were  compressible,  the  lower  strata  would  be  more 
dense  than  the  upper  ones,  and  therefore  the  pressure  would 
increase  at  a  faster  rate  than  the  depth. 

The  following  experiment  will  show  that  the  pressure  of  a 
liquid  upoi^  a  given  base  is  due  to  the  depth  of  the  liquid  and  is 
independent  of  the  volume.  Bend  a  glass  tube  A  B  C,  as  shown 
in  Fig.  132  (a),  and  attach  a  cup  D  E,  into  which  may  be  screwed 


FIG.  132  (a). 


FIG.  132  (6). 


various  shaped  receivers  M,  N,  0,  &c.  Into  the  cup  D  E  pour 
mercury,  which  will  stand  at  a  level,  say  H  D  E.  Now  screw 
into  the  cup  any  one  of  the  receivers,  as  M,  and  pour  in  water  to 
any  desired  height.  The  mercury  will  be  depressed  in  the  cup 
D  E  by  the  pressure  of  the  water,  and  will  rise  in  A  B  to  some 
point  h.  Now  remove  the  vessel  M  and  substitute  in  succession, 
each  of  the  others,  filling  to  the  same  height  as  before.  The  mer- 
cury in  each  case  will  rise  to  the  point  h,  showing  that  the  pres- 
sure upon  the  area  of  mercury  in  the  cup  D  E  is  the  same  in  all 
cases,  for  a  given  height  of  liquid. 

179.  Hydrostatic  Paradox.— To  guard  against  a  possible 
misapprehension  in  this  connection,  the  student  must  be  cau- 
tioned to  distinguish  between  the  pressure  upon  the  bottom  and 
the  weight  of  the  contained  liquid. 

In  the  vessel  A  B  C  D  (Fig.  133),  the  pressure  upon  the  bot- 


120  HYDROSTATICS. 

torn  is  equal  to  the  area  of  the  base  D  C  in  square  inches  multi< 

plied   by  .the  weight  of  a  column  of 

water  of  one  sq.  inch  cross-section  and 

height  E  F,  which  product  is  equal  to 

the  weight  of  a  column  of  cross-section 

D  G  and  height  E  F,   or  the  whole 

volume  m  D  C  n.     But  the  weight  of 

the  contained  water  is  less  than  this, 

as  shown  by  the  figure. 

To  illustrate,  suppose  area  of  base 
D  C  =  12  sq.  inches,  A  D  =  1  inch, 
E  F=  11  inches,  and  x  y  =  1  sq.  inch, 
and  call  the  weight  of  one  cubic  inch  of  water  w.  Then  the 
pressure  upon  the  base  D  C  =  12  x  11  X  w  ==  132  w.  The 
weight  of  the  liquid  =  12  x  1  X  w  +  10  x  1  X  w  =  22  w. 

The  downward  pressure  upon  the  base  is,  as  above,  132  w.  The 
upward  pressure  upon  the  upper  base  A  B  is  equal  to  the  area  of 
the  ring  A  B  multiplied  by  the  height  m-  A,  or  equal  to  11  x 
10  x  w  =  110  w.  Downward  pressure  minus  upward  pressure 
=  weight. 

132  w  —  110  w  =  22  w,  as  before. 

180.  Amount  of  Pressure  in  Water. — One  cubic  foot  of 
water  weighs  1000  ounces,  or  62.5  pounds  avoirdupois.  There- 
fore, the  pressure  on  one  square  foot,  at  the  depth  of  one  foot,  is 
62.5  pounds.  From  this,  as  the  unit  of  hydrostatic  pressure,  it 
is  easy  to  determine  the  pressures  on  all  surfaces,  at  all  depths ; 
for  it  is  obvious  that,  when  the  depth  is  the  same,  the  pressure 
varies  as  the  surface  pressed  upon  ;  and  it  has  been  shown  that, 
on  a  given  surface,  the  pressure  varies  as  the  depth  of  its  centre 
of  gravity ;  it  therefore  varies  as  the  product  of  the  two.  Let 
p  =;  pressure ;  a  =  area  pressed  upon ;  and  d  =  the  depth  of  its 
centre  of  gravity ;  then  p  —  a  d  x  62.5. 

Depth.                         Lbs.  per  sq.  ft.       Depth.                      Lbs.  per  sq.  ft. 
100ft  6,250 


1  ft. 

10   625 

16 1000 


1  mile 330,000 

Smiles 1,650,000 


From  the  above  table  it  may  be  inferred  that  the  pressure  on 
a  square  foot  in  the  deepest  parts  of  the  ocean  must  be  not  far 
from  two  millions  of  pounds  ;  for  the  depth  in  some  places  is 
more  than  five  miles,  and  sea-water  weighs  64.37  pounds,  instead 
of  62.5  pounds.  A  brass  vessel  full  of  air,  containing  only  a  pint, 
and  whose  walls  were  one  inch  thick,  has  been  known  to  be  crushed 
in  by  this  great  pressure,  when  sunk  to  the  bottom  of  the 
ocean. 


HYDROSTATIC    PRESSURE. 


121 


Owing  to  the  increase  of  pressure  FIG.  134. 

with  depth,  there  is  great  difficulty  in 
confining  a  high  column  of  water  by 
artificial  structures.  The  strength  of 
banks,  dams,  flood-gates,  and  aqueduct 
pipes,  must  increase  in  the  same  ratio 
.as  the  perpendicular  depth  from  the 
surface  of  the  water,  without  regard  to 
its  horizontal  extent. 

181.  Column  of  Water  whose 
"Weight  Equals  the  Pressure. — A 

•convenient  mode  of  conceiving  readily 
of  the  amount  of  pressure  on  an  area, 
in  any  given  circumstances,  is  this  : 
•consider  the  area  pressed  upon  to  form 
the  horizontal  base  of  a  hollow  prism  ; 
let  the  height  of  the  prism  equal  the 
average  depth  of  the  area ;  and  then 
suppose  it  filled  with  water.  The 
weight  of  this  column  of  water  is  equal 
to  the  pressure.  For  the  contents  of 
the  prism  (whose  base  =  «,  and  its 
height  =  d),  =  a  d ;  and  the  weight 
of  the  same  =  a  d  x  62.5  Ibs. ;  which 
is  the  same  expression  as  was  obtained 
above  for  the  pressure. 

On  the  bottom  of  a  cubical  vessel 
full  of  water,  the  pressure  equals  the 
weight  of  the  water  ;  on  each  side  of  the  same  the  pressure  is  one- 
half  the  weight  of  the  water ;,  hence,  on  all  the  five  sides  the 
pressure  is  three  times  the  weight  of  the  water ;  and  if  the  top 
were  closed,  on  which  the  pressure  is  zero,  the  pressure  on  the  six 
sides  is  the  same,  three  times  the  weight  of  the  water. 

182.  Illustrations  of  Hydrostatic  Pressure. — A  vessel 
may  be  formed  so  that  both  its  base  and  height  shall  be  great,  but 
its  cubical  contents  small ;  in  which  case,  a  great  pressure  is  pro- 
duced by  a  small  quantity  of  water.     The  hydrostatic  bellows  is 
an  example.     In  Fig.  134  the  weight  which  can  be  sustained  on 
the  lid  D  1  by  the  column  A  D  is  equal  to  that  of  a  prism  or 
cj-linder  of  water,  whose  base  is  D  I,  and  its  height  D  A.    It  is 
immaterial  how  shallow  is  the  stratum  of  water  on  the  base,  or 
how  slender  the  tube  A  D,  if  greater  than  a  capillary  size. 

In  like  manner,  a  cask,  after  being  filled,  may  be  burst  by  an 
additional  pint  of  water  ;  for,  by  screwing  a  long  and  slender  pipe 


122  HYDROSTATICS. 

into  the  top  of  the  cask,  and  filling  it  with  water,  the  pressure  is 
easily  made  greater  than  the  strength  of  the  cask  can  bear. 

183.  Determination  of  Thickness  of  Cylinder.  —  To- 
determine  the  thickness  of  plate  required  in  a  cylindrical  vessel 
that  it  may  sustain  a  given  pressure,  we  assume  that  the  bursting 
results  from  tearing  asunder  the  material  of  the  plate. 

Let  ABE  (Fig.   135)    represent    the    cylindrical  vessel  ? 
E  D  0  A  B  a  longitudinal  sec- 
tion through    the    axis;    put 
a  =  A  B  —  length  in  inches, 

2  r  =  C  D  =  internal  diame-  ^^^^  -^- 

ter  in  inches,  e=A  C=D  E= 
thickness  of  plate  in  inches, 
T  =  tenacity  of  the  material 
in  Ibs.  per  sq.  inch  ;  then 

a  x  e  x  T  =  strength,  or 

resistance  to  tearing  apart,  of  section  B  A  C.  As  there  are  two* 
such  sections  which  resist  the  internal  pressure  the  total  strength 
through  the  section  EDCAB,i&2axexT. 

Call  the  internal  pressure  in  Ibs.  per  sq.  inch  P.  The  total 
bursting  pressure  through  the  section  CD,  acting  upward  and 
downward  to  cause  separation  in  that  plane,  is  equal  to  the  area 
multiplied  by  the  pressure  per  sq.  inch,  or  =  rectangle  2  r  x  a 
X  P.  But  at  the  moment  of  rupture  these  two  must  be  equal, 
therefore 

T   X    P 

2  a  e  T=2rxaxP,  whence  e  =  —  ~  —  which  gives  the 

thickness  when  the  internal  diameter,  the  tenacity  and  the  pres- 
sure are  known.  The  longitudinal  section  through  the  axis  is  the 
weakest  longitudinal  section  that  can  be  taken,  hence  we  need 
consider  no  other. 

To  determine  the  thickness  to  withstand  rupture  through  the 
transverse  section  G  F,  we  have,  area  of  section  of  material 
through  G  F  =  TT  (r  +  0)2  —  -n  r2  =  -n  e  (e  +  2  r),  and  the 
tenacity  of  section  =  ir  e  (e  +  2  r)  T. 

The  bursting-  pressure  upon  the  plane  through  G  F,  exerted 
upon  the  heads  of  the  cylinder,  =  -n  x  r2  x  P.  These  being 
equal  we  -have, 

TT  e  (e  +  2  r}  T  =  -n  r>  P, 


+!    T=rP; 

g  —  ,  whic 
rP 


neglecting  —  ,  which  will  usually  be  a  small  fraction,  we  get 


LEVEL    IN    CONNECTED    VESSELS. 


123 


FIG.  136. 


Comparing  this  with  the  previous  result  we  find  that  the 
transverse  section  requires  only  half  the  thickness  of  material,  for 
a  given  pressure,  which  is  required  by  the  longitudinal  section, 
hence  this  section  need  not  be  considered  in  determining  the 
thickness. 

184.  The  Same  Level  in  Connected  Vessels.— In  tubes 
or  reservoirs  which  communicate  with  each  other,  water  will  rest 
only  when  its  surface  is  at  the 
same  level  in  them  all.  If  water 
is  poured  into  D  (Fig.  136),  it 
will  rise  in  the  vertical  tube  B, 
so  as  to  stand  at  the  same  level 
as  in  D.  For,  the  pressure  to- 
ward the  right  on  any  cross- 
section  E  of  the  horizontal  pipe 
m  n  equals  the  product  of  its 
area  by  its  depth  below  D.  So 
the  pressure  on  the  same  section 
towards  the  left  equals  the  pro- 
duct of  its  area  by  its  depth  be- 
low B.  But  these  pressures  are 
equal,  since  the  liquid  is  at  rest. 

Therefore  E  is  at  equal  depths  below  B  and  D ;  in  other  words, 
B  and  D  are  on  the  same  level.  The  same  reasoning  applies  to- 
the  irregular  tubes  A  and  C,  and  to  any  others,  of  whatever  form 
or  size. 

Water  conveyed  in  aqueducts,  or  running  in  natural  channels 
in  the  earth,  will  rise  just  as  high  as  the  source,  but  no  higher. 

Artesian  wells  illustrate  the  same  tendency  of  water  to  rise  to 
its  level  in  the  different  branches  of  a  tube.  When  a  deep  boring- 
is  made  in  the  earth,  it  may  strike  a  layer  or  channel  of  water 
which  descends  from  elevated  land,  sometimes  very  distant.  The 
pressure  causes  it  to  rise  in  the  tube,  and  often  throws  it  many 
feet  above  the  surface.  Fig.  137  shows  an  artesian  well,  through 

Fio.  137. 


124  HYDROSTATICS. 

which  is  discharged  the  water  that  descends  in  the  porous  stratum 
K  K,  confined  between  the  strata  of  clay  A  B  and  C  D. 

A  tube  driven  to  the  water  bed  anywhere  between  A  or  B  and 
the  lowest  point  in  the  diagram,  might  also  bring  water  to  the 
surface  if  the  flow  below  the  end  of  the  tube  were  sufficiently  ob- 
structed by  friction  ;  hence  an  artesian  well  might  be  successfully 
driven  when  the  inclination  of  the  water  bed  is  wholly  in  one 
direction. 

185.  Centre  of  Pressure. — The  centre  of  pressure  of  any 
surface  immersed  in  water  is  that  point  through  which  passes  the 
resultant  of  all  the  pressures  on  the  surface.     It  is  the  point, 
therefore,  at  which  a  single  force  must  be  applied  in  order  to 
counterbalance  all  the  pressures  exerted  on  the  surface.     If  the 
surface  be  a  plane,  and  horizontal,  the  centre  of  pressure  coincides 
with  the  centre  of  gravity,  because  the  pressures  are  equal  on 
every  part  of  it,  just  as  the  force  of  gravity  is.     But  if  the  plane 
surface  makes  an  angle  with  the  horizon,  the  centre  of  pressure  is 
lower  than  the  centre  of  gravity,  since  the  pressure  increases  with 
the  depth.    For  example ,^if  the  vertical  side  of  a  vessel  full  of 
water  is  rectangular,  the  centre  is  one-third  of  the  distance  from 
the  middle  of  the  base  to  the  middle  of  the  upper  side.     If  tri- 
angular, with  its  lower  side  horizontal,  the  centre  of  pressure  is 
one-fourlh  of  the  distance  from  the  middle  of  the  base  to  the  ver- 
tex.   If  triangular,  with  the  top  horizontal,  the  centre  of  pressure 
is  half  way  up  on  the  bisecting  line. 

[See  Appendix  for  calculations  of  the  place  of  the  centre  of 
pressure.] 

186.  The  Loss  of  Weight  in  Water.— When  a  body  is 
immersed  in  water,  it  suffers  a  pressure  on  every  side,  which  is 
proportional  to  the  depth.     Opposite  components  of  lateral  pres- 
sures, being  exerted  on  surfaces  at  the  same  depth,  balance  each 
other  ;  but  this  cannot  be  true  of  the  vertical  pressures,  since  the 
top  and  bottom  of  the  body  are  at  unequal  depths.     The  upward 
pressure  on  the  bottom  exceeds  the  downward  pressure  on  the 
top ;  and  this  excess  constitutes  the  buoyant  power  of  a  fluid, 
which  causes  a  loss  of  weight. 

A  body  immersed  in  water  loses  weight  equal  to  the  weight  of 
water  displaced.  , 

For  before  the  body  was  immersed,  the  water  occupying  the 
game  space  was  exactly  supported,  being  pressed  upward  more 
than  downward  by  a  force  equal  to  its  own  weight.  The  weight 
of  the  body,  therefore,  is  diminished  by  this  same  difference  of 
pressures,  that  is,  by  the  weight  of  the  displaced  water. 


EQUILIBRIUM    OF    FLOATING    BODIES.  135 

To  show  this  experimentally,  suspend  a  solid  cylinder  A  (Fig. 
138)  below  a  hollow  cylinder  B,  into  which  it  will  fit  with  great 
nicety  ;  attach  both  to  the 

arm  of  a  balance  and  care-  FlG- 138- 

fully  counterpoise  them  ; 
now  pour  water,  or  any 
other  liquid,  into  the  beak- 
er C  until  it  is  full,  and 
the  equilibrium  will  be  de- 
stroyed, the  end  of  the 
beam  D  rising.  Fill  the 
cylinder  B  with  the  same 
liquid,  and  when  it  is  ex- 
actly full,  the  cylinder  A 
will  be  found  to  be  sub- 
merged exactly  to  its  upper  edge,  thus  showing  that  the  buoyancy 
of  the  liquid  in  this  case  is  counteracted  by  a  volume  of  the  same 
liquid  equal  to  the  volume  of  the  submerged  body. 

On  the  supposition  of  the  complete  incompressibility  of  water, 
this  loss  is  the  same  at  all  depths,  because  the  weight  of  displaced 
water  is  the  same.  As  water,  however,  is  slightly  compressible, 
its  buoyant  power  must  increase  a  little  at  great  depths.  Calling 
the  compression  .000046  for  one  atmosphere  (=34  feet  of  water), 
the  bulk  of  water  at  the  depth  of  a  mile  is  reduced  by  about  y^, 
and  its  specific  gravity  increased  in  the  same  ratio ;  so  that, 
possibly*  a  body  might  sink  near  the  surface,  and  float  at  great 
depths  in  the  ocean.  But  this  is  not  probable  in  any  case,  since 
the  same  compressing  force  may  reduce  the  volume  of  the  solid 
as  much  as  that  of  the  water.  And,  furthermore,  the  increase  of 
density  by  increased '  depth  is  so  slow,  that  even  if  solids  were 
incompressible,  most  of  those  which  sink  at  all  would  not  find 
their  floating  place  within  the  greatest  depths  of  the  ocean.  For 
example,  a  stone  twice  as  heavy  as  water  must  sink  100  miles 
before  it  could  float. 

187.  Equilibrium  of  Floating  Bodies.— If  the  body  which 
is  immersed  has  the  same  density  as  water,  it  simply  loses  its 
whole  weight,  and  remains  wherever  it  is  placed.  But  if  it  is  less 
dense  than  water,  the  excess  of  upward  pressure  is  more  than  suf- 
ficient to  support  it ;  it  is,  therefore,  raised  to  the  surface,  and 
comes  to  a  state  of  equilibrium  after  partly  emerging.  In  order 
that  a  floating  body  may  have  a  stable  equilibrium,  the  three  fol- 
lowing conditions  must  be  fulfilled  : 

1.  It  displaces  an  amount  of  water  whose  weight  is  equal  to  its 
otvn. 


126 


HYDROSTATICS. 


2.  Tlie  centre  of  gravity  of  tlie  body  is  in  the  same  vertical  line 
with  that  of  the  displaced  water. 

3.  The  metacenter  is  higher  than  the  centre  of  gravity  of  the 
body. 

The  reason  for  the  first  condition  is  obvious  ;  for  both  the  body 
and  the  water  displaced  by  it  are  sustained  by  the  same  upward 
pressures,  and  therefore  must  be  of  equal  weight. 

FIG.  139. 


That  the  second  is  true,  is  proved  as  follows  :  Let  C  (Fig, 
139,  1)  be  the  centre  of  gravity  of  the  displaced  water,  while  that 
of  the  body  is  at  G.  Now  the  fluid,  previous  to  its  removal,  was 
sustained  by  an  upward  force  equal  to  its  own  weight,  acting 
through  its  centre  of  gravity  C ;  and  the  same  upward  force  now 
acts  upon  the  floating  body  through  the  same  point.  But  the 
body  is  urged  downward  by  gravity  in  the  direction  of  the  vertical 
line  A  G  B.  Were  those  two  forces  exactly  opposite  and  equal, 
they  would  keep  the  body  at  rest ;  but  this  is  the  case  only  when 
the  points  C  and  G  are  in  the  same  vertical  line  ;  in  every  other 
position  of  these  points,  the  two  parallel  forces  tend  to  turn  the- 
body  round  on  a  point  between  them. 

188.  The  Metacenter. — To  understand  the  third  condi- 
tion, the  metacenter  must  be  defined.  A  floating  body  assumes 
a  position  such  that  the  line  through  the  centres  of  gravity  of  the 
body  and  of  the  displaced  water  shall  be  vertical ;  now,  regard  this 
line  so  determined  as  fixed  with  respect  to  the  body,  moving  with 
it  to  any  degree  of  inclination  ;  then  move  the  body  so  that 
this  line  shall  make  an  indefinitely  small  angle  with  its  vertical 
position  ;  the  intersection  of  the  line  as  now  placed  with  the 
vertical  through  the  new  centre  of  gravity  of  the  displaced  water 
is  called  the  metacenter.  When  the  centre  of  gravity  of  the  body 
G  is  lower  than  the  metacenter,  as  in  Fig.  139,  2,  the  parallel 
forces,  downward  through  G  and  upward  through  C,  revolve  the 
body  back  to  its  position  of  equilibrium,  which  is  then  called  a 
stable  equilibrium.  But  if  the  centre  of  gravity  of  the  body  is 


FLOATING    OF    HEAVY    SUBSTANCES.  137 

higher  than  the  metacenter,  as  in  Fig.  139,  3,  the  rotation  is  in 
the  opposite  direction,  and  the  body  is  upset,  the  equilibrium 
being  unstable.  Once  more,  if  the  centre  of  gravity  of  the  body 
is  at  the  metacenter,  the  body  rests  indifferently  in  any  position, 
as,  for  example,  a  sphere  of  uniform  density.  The  equilibrium 
in  this  case  is  called  neutral. 

If  only  the  first  condition  is  fulfilled,  there  is  no  equilibrium  ; 
if  only  the  first  and  second,  the  equilibrium  is  unstable  ;  if  all  the 
three,  the  equilibrium  is  stable. 

In  accordance  with  the  third  condition,  it  is  necessary  to  place 
the  heaviest  parts  of  a  ship's  cargo  in  the  bottom  of  the  vessel, 
and  sometimes,  if  the  cargo  consists  of  light  materials,  to  fill  the 
bottom  with  stone  or  iron,  called  ballast,  lest  the  masts  and  rig- 
ging should  raise  the  centre  of  gravity  too  high  for  stability. 
On  the  same  principle,  those  articles  which  are  prepared  for  life- 
preservers,  in  case  of  shipwreck,  should  be  attached  to  the  upper 
part  of  the  body,  that  the  head  may  be  kept  above  water.  The 
danger  arising  from  several  persons  standing  up  in  a  small  boat 
is  quite  apparent ;  for  the  centre  of  gravity  is  elevated,  and  liable 
to  become  higher  than  the  metacenter,  thus  producing  an  un- 
stable equilibrium. 

189.  Floating  in  a  Small    Quantity  of  Water.— As 

pressure  on  a  given  surface  depends  solely  on  the  depth,  and  not 
at  all  on  the  extent  or  quantity  of  water,  it  follows  that  a  body 
will  float  as  freely  in  a  space  slightly  larger  than  itself  as  on  the 
open  water  of  a  lake.  For  instance,  a  ship  may  be  floated  by  a 
few  hogsheads  of  water  in  a  dock  whose  form  is  adapted  to  it.  In 
such  a  case,  it  cannot  be  literally  true  that  the  displaced  water 
weighs  as  much  as  the  vessel,  when  all  the  water  in  the  dock  may 
not  weigh  a  hundredth  part  as  much.  The  expression  "dis- 
placed water"  means  the  amount  which  would  fill  the  place 
occupied  by  tfhe  immersed  portion  of  the  body.  An  experiment 
illustrative  of  the  above  is,  to  float  a  tumbler  within  another  by 
means  of  a  spoonful  of  water  between. 

190.  Floating  of  Heavy  Substances.— A  body  of  the 
most  dense  material  may  float,  if  it  has  such  a  form  given  it  as  to 
exclude  the  water  from  the  upper  side,  till  the  required  amount 
is  displaced.     Ships  are  built  of  iron,  and  laden  with  substances 
of  greater  specific  gravity  than  water,  and  yet  ride  safely  on  the 
ocean.    A  block  of  any  heavy  material,  as  lead,  may  be  sustained 
by  the  upward  pressure  beneath  it,  provided  the  water  is  excluded 
from  the  upper  side  by  a  tube  fitted  to  it  by  a  water-tight  joint. 

191.  Specific  Gravity. — The  weight  of  a  body  compared 


128  HYDROSTATICS. 

with  the  weight  of  the  same  volume  of  the  standard,  is  called  its 
specific  gravity. 

Distilled  water  at  about  4°  C.  (the  temperature  of  its  greatest 
density)  is  the  standard  for  solids  and  liquids,  and  air,  at  760  mm. 
and  0°,  for  gases.  Therefore  the  specific  gravity  of  a  body  equals 
its  weight  divided  by  the  weight  of  an  equal  volume  of  water  or 
air,  as  the  case  may  be. 

192.  Methods  of  Finding  Specific  Gravity.— 

1.  KEGULAR  SOLID. — Divide  the  iveight  in  grams  by  the  volume  in 
cubic  centimeters. 

Inasmuch  as  a  c.cm.  of  water  weighs  one  gram,  the  volume  of 
a  body  in  c.cm.'s  would  be  the  same  as  the  weight  of  the  same 
volume  of  water. 

2.  SOLED — HEAVIER  THAN  WATEB. — Divide  its  weight  in  air  by  its 
loss  of  weight  in  water. 

A  body  weighed  in  water  is  lighter  by  the  weight  of  an  equal 
volume  of  water. 

3.  SOLID — LIGHTER  THAN  WATER. —  Weigh  the  body  in  air.    Attach 
a  sinker  beneath  the  body  and  make  two  weighings — one  with  the  body 
in  air  and  the  sinker  under  water,  the  other  with  both  body  and  sinker 
under  water.    Divide  the  weiglit  in  air  by  the  difference  of  the  last  two 
iveights. 

A  cork,  with  a  lead  sinker  attached  and  submerged,  would 
weigh  more  than  when  both  cork  and  sinker  were  submerged,  by 
an  amount  equal  to  the  weight  of  the  water  displaced  by  the 
cork. 

In  making  accurate  determinations  of  the  specific  gravity  of 
solids,  care  must  be  taken  to  remove  any  adhering  bubbles  of  air 
before  the  weighing  in  water  is  made. 

If  the  body  whose  specific  gravity  is  required  be  soluble  in 
water,  its  specific  gravity  must  be  determined  with  reference  to 
some  liquid  which  will  not  dissolve  it,  such  as  alcohol,  turpentine, 
a  saturated  solution  of  the  substance  itself,  &c.,  and  then  the 
specific  gravity  so  obtained  must  be  multiplied  by  the  specific 
gravity  of  the  liquid  used,  as  compared  with  water. 

4.  A   LIQUID.  —  Find  the   loss   which   a   body   sustains   weighed 
in  the  liquid  and  then  in  water,  and  divide  the  first  loss  by  the 
second. 

For  the  first  loss  equals  the  weight  of  the  displaced  liquid,  and 
the  second  that  of  the  displaced  water  ;  and  the  volume  in  each 
case  is  the  same,  namely,  that  of  the  body  weighed  in  them. 

But  the  specific  gravity  of  a  liquid  may  be  more  directly  ob- 
tained by  measuring  equal  volumes  of  it  and  of  water  in  a  flask, 


HYDROMETER,  OR  AREOMETER. 


and  finding  the  weight  of  each.     Then  the  weight  of  the  liquid 
divided  by  that  of  the  water  is  the  specific  gravity  required. 

Flasks  for  the  purpose  are  made  with  carefully  ground  stop- 
pers through  which  is  pierced  a  fine  hole  so  that  in  inserting 
the  stopper  there  may  be  an  overflow  through  the  hole,  after  which 
the  flask  having  been  carefully  wiped  off,  it  is  ready  for  weigh- 
ing. 


FIG.  140. 


193.  The  Hydrometer,-  or  Areometer. — In  commerce 
and  the  arts,  the  specific  gravities  of  substances 
are  obtained  in  a  more  direct  and  sufficiently 
accurate  way,  by  instruments  constructed  for 
the  purpose.  The  general  name  for  such  instru- 
ments is  the  hydrometer,  or  areometer.  But 
other  names  are  given  to  such  as  are  limited  to 
particular  uses  ;  as,  for  example,  the  alcoometer 
for  alcohol,  and  the  lactometer  for  milk.  The 
hydrometer,  represented  in  Fig.  140,  consists 
of  a  hollow  ball,  with  a  graduated  stem.  Below 
the  ball  is  a  bulb  containing  mercury,  which 
gives  the  instrument  a  stable  equilibrium  when 
in  an  upright  position.  Since  it  will  descend 
until  it  has  displaced  a  quantity  of  the  fluid 
equal  in  weight  to  itself,  it  will  of  course  sink 
to  a  greater  depth  if  the  fluid  is  lighter.  From  the  depths  to  which 
it  sinks,  therefore,  as  indicated  by  the  graduated  stem,  the  cor- 
responding specific  gravities  are  estimated. 

The  sensibility  of  instruments  of  this 
class  is  increased  by  diminishing  the  diam- 
eter of  the  stem. 

Nicholson's  hydrometer  (Fig.  141)  is  the 
most  useful  of  this  class  of  instruments, 
since  it  may  be  applied  to  finding  the 
specific  gravities  of  solid  as  well  as  liquid 
bodies.  In  addition,  to  the  hollow  ball  of 
the  common  hydrometer,  it  is  furnished 
at  the  top  with  a  pan  A  for  receiving 
weights,  and  a  cavity  beneath  for  holding 
the  substance  under  trial.  The  instru- 
ment is  so  adjusted  that  when  10  grams 
are  placed  in  the  pan,  the  instrument  sinks 
in  distilled  water  at  the  temperature  of 
4°  C.  to  a  fixed  mark,  0,  on  the  stem.  Call- 
ing the  weight  of  the  instrument  W,  the 
weight  of  displaced  water  is  W  +  10. 


FIG.  141. 


130  HYDROSTATICS. 

To  find  the  specific  gravity  of  a  liquid,  place  in  the  pan  such  a 
weight  w  as  will  just  bring  the  mark  to  the  surface.  Then  the 
•weight  of  the  liquid  displaced  is  W  +  w.  But  its  volume  is  equal 
to  that  of  the  displaced  water.  Therefore  its  specific  gravity  is 
W+  w 
W+  10' 

To  find  the  specific  gravity  of  a  solid,  place  in  the  pan  a  frag- 
ment of  it  weighing  less  than  10  grams,  and  add  the  weight  w  re- 
quired to  sink  the  mark  to  the  water-level.  Then  the  weight  of 
the  substance  in  air  is  10  —  w.  Eemove  the  substance  to  the 
-cavity  at  the  bottom  of  the  instrument,  and  add  to  the  weight  in 
the  pan  a  sufficient  number  of  grams  w'  to  sink  the  mark  to 
the  surface.  Then  w'  is  the  loss  of  weight  in  water;  therefore, 

— r—  is  the  specific  gravity  of  the  substance. 

194.  Specific  Gravity  of  Liquids  by  Means  of  Heights. 

— This  method  depends  upon  the  fact  that  the  heights  at  which 
columns  of  liquids  will  be  sustained  by  any  given 
atmospheric  pressure  are  inversely  as  their  specific 
gravities. 

Arrange  two  glass  tubes  A  and  B  (Fig.  142) 
connected  at  the  top  with  a  common  outlet  C,  their 
lower  ends  being  immersed  in  the  liquids  contained 
in  the  beakers  E  and  D.  Exhaust  the  air  by  the 
•outlet  C  till  the  liquids  rise  to  any  desired  height, 
say  to  A  and  B,  and  suppose  the  height  of  the 
column  A,  measured  from  the  surface  of  the  liquid 
in  beaker  E  to  be  one-half  that  of  the  columd  B 
measured  from  the  surface  in  beaker  D ;  then  the 
specific  gravity  of  liquid  E  is  twice  that  of  liquid  D. 

This  method  gives  only  approximate  results, 
depending  upon  the  fineness  of  division  of  the  scales 
used,  corrected  for  capillarity  of  the  tubes. 

195.  Table  of  Specific  Gravities. — An  accurate  knowl- 
edge of  the  specific  gravities  of  bodies  is  important  for  many  pur- 
poses of  science  and  art,  and  they  have  therefore  been  determined 
•with  the  greatest  possible  precision.     The  heaviest  of  ordinary 
substances  is  platinum,  whose  specific  gravity,  when  compressed 
by  rolling,  is  22,  water  being  1 ;    and  the  lightest  is  hydrogen, 
whose  specific  gravity  is  =  .069,  common  air  being  1.     Now,  as 
water  is  about  800  times  as  heavy  as  air,  it  is  (800  -=-  .069  =) 
11,594  times  as  heavy  as  hydrogen.     Therefore  platinum  is  about 
(11,594  x  22  =)  255,068  times  as  heavy  as  hydrogen.     Between 


TABLE     OF     SPECIFIC     GRAVITIES.  131 

these  limits,  1  and  255,068,  there  is  a  wide  range  for  the  specific 
gravities  of  other  substances.  The  values  for  some  substances  are 
given  in  the  following  tables  : 

Water  at  4°  C.  =  1. 

Aluminium 2.6 

Copper 8.5-8.9 

German  silver 8.5 

Glass 2.5-3.5 

Gold 19.3 

Iron  . .    7. 1-7.8 

Lead  11.3 

Mercury 13.6 

Platinum 22.0 

Silver 10. 4 

Wood 0.2-1.2 

Zinc  . . ,  .     7.2 


[At  15°  C.] 

Alcohol 0.7938 

Chloroform 1.499 

Ether 0. 720 

Glycerine 1.260 

Sulphuric  acid  (cone.) • 1.838 


Air  =  1. 

Carbonic  Acid 1.5200 

Hydrogen 0.0692 

Nitrogen 0. 9701 

Oxygen   1.1052 


196.  Floating. — The  human  body,  when  the  lungs  are  filled 
•with  air,  is  lighter  than  water,  and  but  for  the  difficulty  of  keep- 
ing the  lungs  constantly  inflated,  it  would  naturally  float.  With 
a  moderate  degree  of  skill,  therefore,  swimming  becomes  a  very 
easy  process,  especially  in  salt  water.  When,  however,  a  man 
plunges,  as  divers  sometimes  do,  to  a  great  depth,  the  air  in  the 
lungs  becomes  compressed,  and  the  body  does  not  rise  except  by 
muscular  effort.  The  bodies  of  drowned  persons  rise  and  float 
after  a  few  days,  in  consequence  of  the  inflation  occasioned  by 
putrefaction. 

As  rocks  are  generally  not  much  more  than  twice  as  heavy  as 
wat^r,  nearly  half  their  weight  is  sustained  while  they  are  under 
water ;  hence,  their  weight  seems  to  be  greatly  increased  as  soon 
as  they  are  raised  above  the  surface.  It  is  in  part  owing  to  their 
diminished  weight  that  large  masses  of  rock  are  transported  with 


132  HYDROSTATICS. 

great  facility  by  a  torrent.  While  bathing,  a  person's  limbs  feel 
as  if  they  had  nearly  lost  their  weight,  and  when  he  leaves  the 
water,  they  seem  unusually  heavy. 

197.  To  find  the  Magnitude  of  an  Irregular  Body.— 
It  would  be  a  long  and  difficult  operation  to  find  the  exact  con- 
tents of  an  irregular  mineral  by  direct  measurement.  But  it 
might  be  found  with  facility  and  accuracy  by  weighing  it  in  air, 
and  then  finding  its  loss  of  weight  in  water.  The  loss  is  the 
weight  of  a  mass  of  water  having  the  same  volume.  Now,  as 
a  cubic  centimeter  of  water  weighs  one  gram,  if  the  weights  are 
expressed  in  grams,  the  loss  of  weight  equals  the  volume  of  dis- 
placed water  =  the  volume  of  the  mineral. 

198.  Cohesion  and  Adhesion. — What  distinguishes  a 
liquid  from  a  solid  is  not  its  want  of  cohesion  so  much  as  the 
mobility  of  its  particles.  It  is  proved  in  many  ways  that  the  par- 
ticles of  a  liquid  strongly  attract  each  other.  It  is  owing  to  this 
that  water  so  readily  forms  itself  into  drops.  The  same  property 
is  still  more  observable  in  mercury,  which,  when  minutely  divided, 
will  roll  over  surfaces  in  spherical  forms.  When  a  disk  of  almost 
any  substance  is  laid  upon  water,  and  then  raised  gently,  it  lifts  a 
column  of  water  after  it  by  adhesion,  till  at  length  the  edge  of  the 
fluid  begins  to  divide,  and  the  column  is  detached,  not  in  all  parts 
at  once,  but  by  a  successive  rupturing  of  the  lateral  surface.  It 
is  proved  that  the  whole  attraction  of  the  liquid  would  be  far  too 
great  to  be  overcome  by  the  force  applied  to  pull  off  the  disk, 
were  it  not  that  it  is  encountered  by  little  and  little,  at  the  edges 
of  the  column.  But  it  is  the  cohesion  of  the  water  which  is  over- 
come in  this  experiment ;  for  the  upper  lamina  still  adheres  to  the 
disk.  By  a  pair  of  scales  we  find  that  it  requires  the  same  force 
to  draw  off  disks  of  a  given  size,  whatever  the  materials  may  be, 
provided  they  are  wet  when  detached.  This  is  what  might  be 
expected,  since  in  each  case  we  break  the  attraction  between  two 
laminae  of  water.  But  if  we  use  disks  which  are  not  wet  by  the 
liquid,  it  is  not  generally  true  that  those  of  different  material  will 
be  removed  by  the  same  force ;  indicating  that  some  substances 
adhere  to  a  given  liquid  more  strongly  than  others. 

These  molecular  attractions  extend  to  an  exceedingly  small 
distance,  as  is  proved  by  many  facts.  A  lamina  of  water  adheres 
as  strongly  to  the  thinnest  disk  that  can  be  used  as  to  a  thick  one ; 
so,  also,  the  upper  lamina  coheres  with  equal  force  to  the  next 
below  it,  whether  the  layer  be  deep  or  shallow. 

\        199.  Capillary  Action. — This  name  is  given  to  the  molec- 
ular forces,  adhesion  and  cohesion,  when  they  produce  disturbing 


CAPILLARY    ACTION.  133 

Affects  on  the  surface  of  a  liquid,  elevating  it  above  or  depressing 
it  below  the  general  level.  These  effects  are  called  capillary,  be- 
cause most  strikingly  exhibited  in  very  fine  (hair-sized]  tubes. 

The  liquid  will  be  elevated  in  a  concave  curve,  or  depressed  in  a 
convex  curve,  by  the  side  of  the  solid,  according  as  the  attraction 
of  the  liquid  molecules  for  each  other  is  less  or  greater  than  tivice 
the  attraction  between  the  liquid  and  the  solid. 


143. 


Case  1st.  Let  H  K  (Fig.  143,  1)  and  L  M  be  a  section  of  the 
vertical  side  of  a  solid,  and  of  the  general  level  of  the  liquid.  The 
particle  A,  where  these  lines  meet,  is  attracted  (so  far  as  this  sec- 
tion is  concerned)  by  all  the  particles  of  an  insensibly  small  quad- 
rant of  the  liquid,  the  resultant  of  which  attractions  is  in  the  line 
A  D,  45°  below  A  M.  It  is  also  attracted  by  all  the  particles  in 
two  quadrants  of  the  solid,  and  the  resultants  are  in  the  directions 
A  B,  45°  above,  and  A  B',  45°  below  L  M. 

Now  suppose  the  force  A  D  to  be  less  than  twice  A  B  or  A  B'. 
Cut  off  CD  =  A  B ;  tj^en  A  B,  being  opposite  and  equal  to  CD, 
is  in  equilibrium  with  it.  The  remainder  A  C,  being  less  than 
A  B',  their  resultant  A  E  will  be  directed  toward  the  solid  ;  and 
therefore  the  surface  of  the  liquid,  since  it  must  be  perpendicular 
to  the  resultant  of  forces  acting  on  it  (Art.  175),  takes  the  direc- 
tion represented  ;  that  is,  concave  upward. 

Case  2d.  Let  A  D  (Fig.  143,  2),  the  attraction  of  A  toward 
the  liquid  particles,  be  more  than  tivice  A  B,  the  attraction  toward 
a  quadrant  of  the  solid.  Then,  making  CD  equal  to  A  B,  these 
two  resultants  balance  as  before  ;  and  as  A  C  is  greater  than  A  B', 
the  angle  between  A  C  and  the  resultant  A  E  is  less  than  45°, 
and  A  is  drawn  away  from  the  solid.  Therefore  the  surface, 
being  perpendicular  to  the  resultant  of  the  molecular  forces  acting 
on  it,  is  convex  upward. 

Case  3d.  If  A  D  (Fig.  143,  3)  be  exactly  twice  A  B,  then  CD 
balances  A  B,  and  the  resultant  of  A  C  and  A  B'  is  A  E  in  a  ver- 
tical direction  ;  therefore  the  surface  at  A  is  level,  being  neither 
elevated  nor  depressed. 

Case  1st  occurs  whenever  a  liquid  readily  wets  a  solid,  if 
brought  in  contact  with  it,  as,  for  example,  water  and  clean  glass. 


134 


HYDROSTATICS. 


Case  2d  occurs  when  a  solid  cannot  be  ^vet  by  a  liquid,  as  glass 
and  mercury.  Case  3d  is  rare,  and  occurs  at  the  limit  between 
the  other  two ;  water  and  steel  afford  as  good  an  example  as  any. 

200.  Capillary  Tubes. — In  fine  tubes  these  molecular  forces, 
affect  the  entire  columns  as  well  as  their  edges.    If  the  material 
of  the  tube  can  be  wet  by  a  liquid,  it  will  raise  a  column  of  that 
liquid  above  the  level,  at  the  same  time  making  the  top  of  the 
column  concave.     If  it  is  not  capable  of  being  wet,  the  liquid  is 
depressed,  and  the  top  of  the  column  is  convex.     The  first  case  is 
illustrated  by  glass  and  water  ;  the  second  by  glass  and  mercury. 

The  materials  being  given,  the  distance  by  which  the  liquid  is 
elevated  or  depressed  varies  inversely  as  the  diameter.  Therefore 
the  product  of  the  two  is  constant. 

The  amount  of  elevation  and  depression  depends  on  the 
strength  of  the  molecular  forces,  rather  than  on  the  specific 
gravity  of  the  liquids.  Alcohol,  though  lighter  than  water,  is. 
raised  only  half  as  high  in  a  glass  tube. 

201.  Parallel   and    Inclined    Plates.— Between  parallel 
plates  a  liquid  rises  or  falls  half  as  far  as  in  a  tube  of  the  same 
diameter.     This  is  because  the  sustaining  force  acts  only  on  two 
sides  of  each  filament,  while  in  a  tube  it  acts  on  all  sides.     There- 
fore, as  in  tubes  the  height  varies  inversely  as  the  diameter,  so  in 
plates  the  height  varies  inversely  as  the  distance  between  them. 

If  the  plates  are  inclined  to  each  other,  having  their  edge  of 
meeting  perpendicular  to  the  horizon,  the  surface  of  a  liquid  rising 
between  them  assumes  the  form  of  a  hyperbola,  whose  brandies 
approach  the  vertical  edge,  and  the  water-level,  as  the  asymptotes 
of  the  curve.  This  results  from  the  law  already  stated,  that  the 
height  varies  inversely  as  the  distance  between  the  plates.  Let 
the  edge  of  meeting,  A  H(F\g,  144), 
be  the  axis  of  ordinates,  and  the 
line  in  which  the  level  surface  of 
the  water  intersects  the  glass,  A  P, 
the  axis  of  abscissas.  Let  B  C, 
D  E,  be  any  ordinates,  and  A  B, 
A  D,  their  abscissas,  and  B  L.,  D  K, 
the  distances  between  the  plates. 
By  the  law  of  capillarity,  the  heights 
BC,DE,  are  inversely  asBL,D  K. 
But,  by  the  similar  triangles,  A  B  L, 
A  D  K,  B  L,  D  K,  are  as  AB,AD  ; 
therefore,  B  C,  D  E,  are  inversely  as  A  B,  A  D ;  and  this  is  a 
property  of  the  hyperbola  with  reference  to  the  centre  and  asymp- 
totes, that  the  ordinates  are  inversely  as  the  abscissas. 


FIG.  144. 


ILLUSTRATIONS    OF     CAPILLARITY.  135 

202.  Effects  of  Capillarity  on  Floating  Bodies.— Some 
cases  of  apparent  attractions   and   repulsions  between   floating 
bodies  are  caused  by  the  forms  which  the  liquid  assumes  on  the 
sides  of  the  bodies.     If  two  balls  raise  the  water  about  them,  and 
are  so  near  to  each  other  that  the  concave  surfaces  between  them 
meet  in  one,  they  immediately  approach  each   other  till   they 
touch;   and  then,  if  either  be  moved,  the  other  will  follow  it. 
The  water,  which  is  raised  and  hangs  suspended  between  them, 
draws  them  together. 

Again,  if  each  ball  depresses  the  water  around  it,  they  will  also 
move  to  each  other,  and  be  held  together,  so  soon  as  they  are 
near  enough  for  the  convex  surfaces  to  meet.  In  this  case,  they 
.are  not  pulled,  but  pushed  together  by  the  hydrostatic  pressure 
•of  the  higher  water  on  the  outside. 

Once  more,  if  one  ball  raises  the  water,  and  the  other  depresses 
it,  and  they  are  brought  so  near  each  other  that  the  curves  meet, 
they  immediately  move  apart,  as  if  repelled.  For  now  the  equi- 
librium is  destroyed  in  a  way  just  the  reverse  of  the  preceding 
cases.  The  water  between  the  balls  is  too  high  for  that  which  de- 
presses, and  too  low  for  that  which  raises  the  water,  so  that  the 
former  is  poshed  away,  and  the  latter  is  drawn  away. 

The  first  case,  which  is  by  far  the  most  common,  explains  the 
fact  often  observed,  that  floating  fragments  are  liable  to  be 
.gathered  into  clusters ;  for  most  substances  are  capable  of  being 
wet,  and  therefore  they  raise  the  water  about  them. 

203.  Illustrations  of  Capillary  Action.— It  is  by  capil- 
lary action  that  a  part  of  the  water  which  falls  on  the  earth  is 
kept  near  its  surface,  instead  of  sinking  to  the  lowest  depths  of 
the  soil.     This  force  aids  the  ascent  of  sap  in  the  pores  of  plants. 
It  lifts  the  oil  between  the  fibres  of  the  lamp-wick  to  the  place  of 
combustion.     Cloth  rapidly  imbibes  moisture  by  its  numerous 
capillary  spaces,  so  that  it  can  be  used  for  wiping  things  dry.     If 
paper  is  not  sized,  it  also  imbibes  moisture  quickly,  and  can  be 
used  as  blot timj -paper  ;  but  when- its  pores  are  filled  with  sizing, 
to  fit  it  for  writing,    it  absorbs  moisture  only  in  a  slight  de- 
gree and  the  ink  which  is  applied  'to  it  must  dry  by  evapora- 
tion. 

The  great  strength  of  the  capillary  force  is  shown  in  the  effects 
produced  by  the  swelling  of  wood  and  other  substances  when  kept 
"wet.  Dry  wooden  wedges,  driven  into  a  groove  cut  around  a 
cylinder  of  stone,  and  then  occasionally  wet,  will  at  length  cause 
it  to  break  asunder.  As  the  pores  between  the  fibres  of  a  rope 
run  around  it  in  spiral  lines,  the  swelling  of  a  rope  caused  by 
keeping  it  wet  will  contract  its  length  with  immense  force. 


136  HYDROSTATICS. 

204.  Questions  in  Hydrostatics. — 

1.  The  diameters  of  the  two  cylinders  of  a  hydraulic  press  are 
one  inch  and  one  foot,  respectively  ;  before  the  piston  descends,  the 
column  of  water  in  the  small  cylinder  is  two  feet  higher  than  the 
bottom  of  the  large  piston.     Suppose  that  by  a  screw  a  force  of 
500  Ibs.  is  applied  to  the  small  piston ;  what  is  the  whole  force 
exerted  on  the  large  piston  at  the  beginning  of  the  stroke  ? 

Ans.  72098.17  Ibs. 

2.  A  junk  bottle,  whose  lateral   surface  contained  50  square 
inches,  being  let  down  into  the  sea  3000  feet,  what  pressure  do- 
the  sides  of  the  bottle  sustain,  a  cubic  foot  of  sea  water  weighing 
64.37  Ibs.?  Ans.  67052.08  +  lbs. 

3.  What  will  be  the  apparent  weight  in  water  of  a  piece  of 
rock-crystal  (density  2.7)  which  weighs  35  grams  in  vacuo? 

Ans.  22.04  grams. 

4.  A  bar  of  aluminium  (density  2.6)  weighs  54.8  grams  in  vacuo  : 
what  will  be  the  loss  of  weight  when  it  is  weighed  in  water  ? 

Ans.  21.08  grams. 

5.  An  irregular  solid  is  found  to  weigh  98  grams  in  vacuo  and 
64  grams  in  water :  what  is  its  volume  ? 

6.  A  solid  cube,  4  inches  in  the  side,  is  formed  of  a  substance 
of  specific  gravity  12.5  :  what  will  its  apparent  weight  in  water 
be? 

7.  A  body  which  weighs  24  grams  in  air  is  found  to  weigh  20 
grams  in  water:  what  will  be  its  apparent  weight  in  alcohol  of 
specific  gravity  0.8?  Ans.  20.8  grams. 

8.  A  body  which  weighs  35  grams  in  air  is  found  to  weigh  30 
grams  in  one  fluid  and   25  grams  in  another:,  what  will  be  its 
weight  when  immersed  in  a  mixture  containing  equal  volumes  of 
the  two  fluids?  -4ns.  27.5  grams. 

9.  Two  bodies  are  in  equilibrium   when  suspended  in   water 
from  the  arms  of  a  balance  :  the  mass  of  the  one  body  is  28  and  its 
density  is  5.6  ;  if  the  mass  of  the  other  is  36,  what  is  its  density? 

-4ns.  2.77. 

10.  A  specific  gravity  bottle  weighs  14.72  grams  when  empty, 
39.74  grams  when  filled  with  water,  and  44.85  gi-ams  when  filled 
with  a  solution  of  common  salt :  what  is  the  specific  gravity  of  the 
solution?  Ans.  1.204. 

11.  A  Nicholson's  hydrometer,  when  floating  in  water,  required 
a  weight  of  0.15  grams  to  be  placed  upon  the  upper  pan  in  order 
to  make  it  sink  to  a  fixed  mark  on  the  stem  ;  and  5.72  grams  had 
to  be  placed  upon  the  pan  in  order  to  make  it  sink  to  the  same 
mark  in  a  solution  of  salt.     If  the  hydrometer  weighed  94.47  grams,, 
what  was  the  specific  gravity  of  the  solution? 


VELOCITY     OF     DISCHARGE.  137 

12.  The  specific  gravity  of  lead  being  11.35  ;  of  cork,  .24 ;  of 
fir,  .45  ;  how  much  cork  must  be  added  to  60  Ibs.  of  lead,  that  the 
united  mass  may  weigh  as  much  as  an  equal  bulk  of  fir  ? 

Am.  65.8527  Ibs. 


CHAPTER    II. 

HYDRODYNAMICS. 

205.  Depth  and  Velocity  of  Discharge.— From  an  aper- 
ture which  is  small,  compared  with  the  breadth  of  the  reservoir, 
the  velocity  of  discharge  varies  as  the  square  root  of  the  depth.  For 
the  pressure  on  a  given  area  varies  as  the  depth  (Art.  178).  If  the 
surface  be  removed,  this  pressure  becomes  a  force  which  is  measured 
by  the  momentum  of  the  water  ;  therefore  the  momentum  varies  as 
the  depth  (d).  But  momentum  varies  as  the  mass  (m)  multiplied  by 
the  velocity  (v) ;  hence  m  v  oc  d.  But  it  is  obvious  that  m  and  v 
vary  alike,  since  the  greater  the  velocity,  the  greater  in  the  same 

ratio  is  the  quantity  discharged.     Therefore,  ?n2  <x  d,  or  m  oc  d*  ; 

also  v2  oc   d,  or  v  oc  d?. 

Not  only  does  the  velocity  vary  as  the  square  root  of  the  depth 
of  the  orifice,  but  it  is  equal  to  that  acquired  by  a  body  falling 
through  the  depth. 

Let  h  —  the  height  of  the  liquid  above  the  orifice,  and  A'  =  the 
height  of  an  infinitely  thin  layer  at  the  orifice. 

If  this  thin  layer  were  to  fall  through  the  height  h',  under  the 
action  of  its  own  weight  or  pressure,  the  velocity  acquired  would 
be  v  =  V%gV  (Art,  27). 

Denoting  the  velocity  generated  by  the  pressure  of  the  entire 
column  by  v,  we  have,  since  velocity  oc  V  depth, 
v  :  v'  : :  Vh  :  Vh',  or 
v  :  V%  g  h'  : :  ^/h  :  V  h' ; 
.-.  v  =  V%  g  h. 

But  V  %gh  is  also  the  velocity  acquired  in  falling  through  the 
distance  h  (Art.  27). 

From  an  orifice  16.1  feet  below  the  surface  of  water,  the  veloc- 
ity of  discharge  is  32.2  feet  per  second,  because  this  is  the  velocity 
acquired  in  falling  1G.1  feet ;  and  at  a  depth  four  times  as  great. 


138  HYDRODYNAMICS. 

that  is,  64.4  feet,  the  velocity  will  only  be  doubled,  that  is,  64.4  feet 
per  second. 

As  the  velocity  of  discharge  at  any  depth  is  equal  to  that  of  a 
body  which  has  fallen  a  distance  equal  to  the  depth,  it  is  theoreti- 
cally immaterial  whether  water  is  taken  upon  a  wheel  from  a  gate 
at  the  same  level,  or  allowed  to  fall  on  the  wheel  from  the  top 
of  the  reservoir.  In  practice,  however,  the  former  is  best,  on 
account  of  the  resistance  which  water  meets  with  in  falling 
through  the  air. 

206.  Descent  of  Surface. — When  water  is  discharged  from 
the  bottom  of  a  cylindric  or  prismatic  vessel,  the  surface  descends 
with  a  uniformly  retarded  motion.     For  the  velocity  with  which 
the  surface  descends  varies  as  the  velocity  of  the  stream,  and 
therefore  as  the  square- root  of  the  depth  (Art.  205).     But  this  is  a 
characteristic  of  uniformly  retarded  motion,   that  the  velocity 
varies  as  the  square  root  of  the  distance  from  the  point  where  the 
motion  terminates,  as  in  the  case  of  a  body  ascending  perpendicu- 
larly from  the  earth. 

The  descent  of  the  surface  of  water  in  a  prismatic  vessel  has 
been  used  for  measuring  time.  The  clepsydra,  or  water-clock 
of  the  Eomans,  was  a  time-keeper  of  this  description.  The  grad- 
uation must  increase  upAvard,  as  the  odd  numbers  1,  3,  5,  7,  &c. ; 
since,  by  the  law  of  this  kind  of  motion,  the  spaces  passed  over  in 
equal  times  are  as  those  numbers. 

If  a  prismatic  vessel  is  kept  full,  it  discharges  twice  as  much 
water  in  the  same  time  as  if  it  is  allowed  to  empty  itself.  For  the 
velocity  in  the  first  instance,  is  uniform  ;  and  in  the  second  it  is 
uniformly  retarded,  till  it  becomes  zero.  We  reason  in  this  case, 
therefore,  as  in  regard  to  bodies  moving  uniformly,  and  with 
motion  uniformly  accelerated  from  rest,  or  uniformly  retarded 
till  it  ceases  (Art.  21),  that  the  former  motion  is  twice  as  great  as 
the  latter. 

207.  Discharge  from  Orifices  in  Different  Situations. 
— Other  circumstances  besides  area  and  depth  of  the  aperture  are 
found  to  have  considerable  influence  on  the  velocity  of  discharge. 
Observations  on  the  directions  of  the  filaments  are  made  by  intro- 
ducing into  the  water  particles  of  some  opaque  substance,  having 
the  same  density  as  water,  whose  movements  are  visible.     From 
such  observations  it  appears  that  the  particles  of  water  descend  in 
vertical  lines,  until  they  arrive  within  three  or  four  inches  of  the 
aperture,  when  they  gradually  turn  in  a  direction  more  or  less 
oblique  toward  the  place  of  discharge.     This  convergence  of  the 
filaments  extends  outside  of  the  vessel,  and  causes  the  stream  to 


DISCHARGE    FROM    ORIFICES. 


139 


diminish  for  a  short  distance,  and  then  increase.  The  smallest 
section  of  the  stream,  called  the  vena  contracta,  is  at  a  distance 
from  the  aperture  varying  from  one-half  of  its  diameter  to  the 
whole. 

If  water  is  discharged  through  a  circular  aperture  in  a  thin 
plate  in  the  bottom  of  the  reservoir,  and  at  a  distance  from  the 
sides,  as  in  Fig.  145,  1,  the  filaments  form  the  vena  contracta  at  a 
distance  beyond  the  aperture  equal  to  one-half  of  its  diameter ;  the 
area  of  the  section  at  the  vena  contracta  is  less  than  two-thirds 
(0.64)  of  the  area  of  the  aperture  ;  this  contraction  also  lessens 
the  theoretical  velocity  by  about  four  per  cent.,  leaving  .96  v  for 
the  final  velocity  ;  combining  these  two  causes,  it  is  found  that  for 
circular  orifices  of  £  to  6  inches  in  diameter,  with  from  four  to 
20  feet  head  of  water,  the  actual  discharge  is  only  .615  of  the 
theoretical  discharge. 

FIG.  145. 

3  4 


If  the  reservoir  terminates  in  a  short  pipe  or  ajutage,  whose 
interior  is  adapted  to  the  curvature  of  the  filaments,  as  far  as  to 
the  vena  contracta,  or  a  little  beyond,  as  in  Fig.  145, 2,  it  is  found 
the  most  favorable  for  free  discharge,  which  in  some  cases  reaches 
0.98  of  the  theoretical  discharge.  The  stream  is  smooth  and  pel- 
lucid like  a  rod  of  glass.  The  most  unfavorable  form  is  that  in 
which  the  ajutage,  instead  of  being  external,  as  in  the  case  just 
described,  projects  inward,  as  in  Fig.  145,  3  ;  the  filaments  in  this 
case  reach  the  aperture,  some  ascending,  others  descending,  and 
therefore  interfere  with  each  other.  Hence  the  stream  is  much 
roughened  in  its  appearance,  and  the  flow  is  only  0.53  of  what  is 
due  to  the  size  of  the  aperture  and  its  depth. 

When  the  aperture  is  through  a  thin  plate,  the  contraction  of 
the  stream  and  the  amount  of  discharge  are  both  modified  by  the 
circumstance  of  being  near  one  or  more  sides  of  the  reservoir. 
There  is  little  or  no  contraction  on  the  side  next  the  wall  of  the 
vessel,  since  the  filaments  have  no  obliquity  on  that  side  ;  and  the 
quantity  is  on  that  account  increased.  The  filaments  from  the 
opposite  side  also  divert  the  stream  a  few  degrees  from  the  per- 
pendicular (Fig.  145,  4). 


140  HYDRODYNAMICS. 

208.  Friction  in  Pipes. — As  has  just  been  stated,  an  ajutage 
extending  to  or  slightly  beyond  the  vena  contracta,  and  adapted 
to  the  form  of  the  stream,  very  much  increases  the  quantity  dis- 
charged ;  but  beyond  that,  the  longer  the  pipe,  the  more  does  it 
impede  the  discharge  by  friction.     For  a  given  quantity  of  water 
flowing  through  a  pipe  the  resistance  of  friction  increases  with 
the  number  of  points  with  which  the  water  comes  in  contact; 
that  is,  the  resistance  is  in  proportion  to  the  wetted  surface  ;  for 
every  particle  of  water  in  contact  with  the  interior  surface  of  the 
pipe,  acts  as  a  retarding  force.      Now  let  f  be  the  resistance 
of  friction  in  a  pipe   of  unit  diameter,  length   and  velocity; 
then  the  resistance  in  a  pipe  I  feet  long  and  d  feet  in  diameter 
with  a  unit  of  velocity  will  befdl;  but  the  quantity  of  water 
delivered  by  this  pipe  will  be  d2  times  that  delivered  by  the 
former,  in  unit  of  time  with  same  velocity,  since  areas  of  cross- 
sections  are  to  each  other  as  squares  of  their  diameters;  there- 
fore for  the  same  quantity  of  water  delivered,  the  resistance  of 

friction  in  the  latter  pipe  will  be  ^        or  —^  that  is  to  say, 

the  resistance  of  friction  in  pipes  is  directly  as  their  lengths  and 
inversely  as  their  diameters,  the  velocity  being  constant.  In  order, 
therefore,  to  convey  water  at  a  given  rate  through  a  long  pipe,  it 
is  necessary  either  to  increase  the  head  of  water  or  to  enlarge  the 
pipe,  so  as  to  compensate  for  friction. 

An  aqueduct  should  be  as  straight  as  possible,  not  only  to- 
avoid  unnecessary  increase  of  length,  but  because  the  force  of  the 
stream  is  diminished  by  all  changes  of  direction.  '  If  there  must 
be  change,  it  should  be  a  gradual  curve,  and  not  an  abrupt  turn. 
When  a  pipe  changes  its  direction  by  an  angle,  instead  of  a  curve, 
there  is  a  useless  expenditure  of  force ;  a  change  of  90°  requires 
that  the  head  of  water  should  be  increased  by  nearly  the  height  due 
to  the  velocity  of  discharge.  For  instance,  if  the  discharge  is  eight 
feet  per  second  (which  is  the  velocity  due  to  one  foot  of  fall),  then 
a  right  angle  in  the  pipe  requires  that  the  head  of  water  should  be 
increased  by  nearly  one  foot,  in  order  to  maintain  that  velocity. 

Empirical  formulae,  based  upon  the  results  of  experiments  for 
the  velocity  of  flow  in  pipes,  and  for  the  loss  of  head  due  to  bends 
and  angles  in  the  pipe,  are  given  in  works  treating  of  Practical 
Hydraulics.  The  derivation  and  development  of  such  formulae  is 
beyond  the  scope  of  a  work  like  this. 

209.  Jets. — Since  a  body,  when  projected  upward  with  a  cer- 
tain velocity,  will  rise  to  the  same  height  as  that  from  which  it 
must  have  fallen  to  acquire  that  velocity,  therefore,  if  water  issue 
from  the  side  of  a  vessel  through  a  pipe  bent  upward,  it  would, 


RIVERS. 


141 


were  it  not  for  the  resistance  of  the  air  and  friction  at  the  orifice,, 
rise  to  the  level  of  the  water  in  the  reservoir.  If  water  is  dis- 
charged from  an  orifice  in  any  other  than  a  vertical  direction,  it 
describes  a  parabola,  since  each  particle  may  be  regarded  as  a  pro- 
jectile (Art.  47). 

If  a  semicircle  be  described  on  the  perpendicular  side  of  a 
vessel  as  a  diameter,  and  water 
issue  horizontally  from  any  point, 
its  range,  measured  on  the  level  of 
the  base,  equals  tunce  the  ordinate 
of  that  point.  For,  the  velocity 
with  which  the  fluid  issues  from 
the  vessel,  being  that  which  is  due 
to  the  height  B  G  (Fig.  146),  is 
V%  g  •  B  G  (Art  27).  But  after 
leaving  the  orifice,  it  arrives  at  the 
horizontal  plane  in  the  time  in  which  a  body  would  fall  freely 

through  G  D,  which  is  |/ j    <L?e     Since  the  horizontal  motion. 

y 
equals  the  product  of  the  time  by  the  veloc- 

~%rcTD 


is  uniform, 

ity  ;  that  is,  D  E  = 


G  .  G  D= 


2  G  H,  or  twice  the  ordinate  of  the  semicircle  at  the  place  of  dis- 
charge. 

The  greatest  range  occurs  when  the  fluid  issues  from  the 
centre,  for  then  the  ordinate  is  greatest ;  and  the  range  at  equal 
distances  above  and  below  the  centre  is  the  same. 

The  remarks  already  made  respecting  pipes  apply  to  those 
which  convey  water  to  the  jets  of  fire-engines  and  fountains.  If 
the  pipe  or  hose  is  very  long,  or  narrow,  or  crooked,  or  if  the  jet- 
pipe  is  not  smoothly  tapered  from  the  full  diameter  of  the  hose  to 
the  aperture,  much  force  is  lost  by  friction  and  other  resistances, 
especially  in  great  velocities.  If  the  length  of  hose  is  even  tiventy 
times  as  great  as  its  diameter,  32  per  cent,  of  height  is  lost  in  the 
jet,  and  more  still  when  the  ratio  of  length  to  diameter  is  greater 
than  this. 

210.  Rivers, — Friction  and  change  of  direction  have  great 
influence  on  the  flow  of  rivers.  A  dynamical  equilibrium,  as  it  is 
called,  exists  between  gravity,  which  causes  the  descent,  and  the 
resistances,  which  prevent  acceleration  at  any  given  point,  beyond 
a  certain  moderate  limit ;  as  the  same  quantity  of  water  must  pass 
every  cross  section  of  the  stream  in  the  same  unit  of  time,  under 
ordinary  conditions,  the  velocity  varies  inversely  as  the  area  of 


142 


HYDRODYNAMICS. 


the  cross  section.  The  velocity  in  all  parts  of  the  same  section, 
however,  is  not  the  same  ;  it  is  greatest  at  that  part  of  the  surface 
where  the  depth  is  greatest,  and  least  in  contact  with  the  bed  of 
the  stream. 

To  find  the  mean  velocity  through  a  given  section,  it  is  neces- 
sary to  float  bodies  at  various  places  on  the  surface,  and  also  below 
it,  to  the  bottom,  aud  to  divide  the  sum  of  all  the  velocities  thus 
obtained,  by  the  number  of  observations.  To  obtain  the  quantity 
of  water  which  flows  through  a  given  section  of  a  river,  having 
determined  the  velocity  as  above,  find  next  the  area  of  the  section, 
by  taking  the  depth  at  various  points  of  it,  and  multiplying  the 
mean  depth  by  the  breadth.  The  quantity  of  water  is  then  found 
by  multiplying  the  area  by  the  velocity. 

The  increased  velocity  of  a  stream  during  a  freshet,  while  the 
stream  is  confined  within  its  banks,  exhibits  something  of  the  ac- 
celeration which  belongs  to  bodies  descending  on  an  inclined 
plane.  It  presents  the  case  of  a  river  flowing  upon  the  top  of 
another  river,  and  consequently  meeting  with  much  less  resistance 
than  when  it  runs  upon  the  rough  surface  of  the  earth  itself.  The 
augmented  force  of  a  stream  in  a  freshet  arises  from  the  simulta- 
neous increase  of  the  quantity  of  water  and  the  velocity.  In  con- 
sequence of  the  friction  of  the  banks  and  beds  of  rivers,  and  the 
numerous  obstacles  they  meet  with  in  their  winding  course,  their 
velocity  is  usually  very  small,  not  more  than  three  or  four  miles 
per  hour;  whereas,  were  it  not  for  these 
impediments,  it  would  become  immensely 
great,  and  its  effects  would  be  exceedingly 
disastrous.  A  very  slight  declivity  is  suffi- 
cient for  giving  the  running  motion  to 
water.  The  largest  rivers  in  the  world 
fall  about  five  or  six  inches  in  a  mile. 

211.  Hydraulic  Pumps.— The  most 
*  common  pumps  for  raising  water  operate 
on  a  principle  of  pneumatics,  and  will  be  de- 
scribed under  that  subject. 

In  the  lifting  pump  the  water  is  pushed 
tip  in  the  pump  tube  by  a  piston  placed  be- 
low the  water-level.  In  the  tube  A  B  (Fig. 
147)  is  a  fixed  valve  V,  a  little  below  the 
water-level  L  L,  while  still  lower  is  the  pis- 
ton P,  in  which  there  is  a  valve.  Both  of 
these  valves  open  upward.  The  piston  is 
attached  to  a  rod,  which  extends  downward 
to  the  frame  F F.  This  frame  can  be  moved 


FIG.  147. 


CENTRIFUGAL    PUMPS.  143 

up  and  down  on  the  outside  of  the  tube  by  a  lever.  When  the 
piston  descends,  the  water  passes  through  its  valve  by  hydrostatic 
pressure ;  and  when  raised,  it  pushes  the  water  before  it  through 
the  fixed  valve,  which  then  prevents  its  return.  In  this  manner, 
by  repeated  strokes,  the  water  can  be  driven  to  any  height  which 
the  instrument  can  bear. 

The  chain  pump  consists  of  an  endless  chain  with  circular 
disks  attached  to  it  at  intervals  of  a  few  inches,  which  raise  the 
water  before  them  in  a  tube,  by  means  of  a  wheel  over  which  the 
chain  passes;  the  wheel  may  be  turned  by  a  crank.  The  disks 
cannot  fit  closely  in  the  tube  without  causing  too  great  resistance; 
hence,  a  certain  velocity  is  requisite  in  order  to  raise  water  to  the 
place  of  discharge  ;  and  after  the  working  of  the  pump  ceases,  the 
water  soon  descends  to  the  level  in  the  well. 

212.  Centrifugal     Pumps. — Water    may    also    be    raised 
through  small  heights  and  in  great  volume  by  the  centrifugal 
pump.    This  consists  of  revolving  curved,  hollow  arms,  connected 
with  a  hollow  axis  through  which  the  water  enters.     As  this  axis 
is  made  to  rotate  in  a  direction  contrary  to  the  curvature  of  the 
arms  the  centrifugal  force  causes  the  water  to  leave  the  arms  and 
move  off  in  tangents  ;  a  casing  drum  inclosing  the  revolving  por- 
tion forces  the  water  to  move  around  in  a  vortex  till  it  reaches  a 
delivery  pipe  entering  the  drum  as  a  tangent,  through  which  it  is 
discharged.     A  high  delivery  requires  so  great  velocity  that  the 
pump  becomes  inferior  in  efficiency  to  other  forms. 

213.  The  Hydraulic    Ram. — When  a  large  quantity  of 
water  is  descending  through  an  inclined  pipe,  if  the  lower  extrem- 
ity is  suddenly  closed,  since  water  is  nearly  incompressible,  the 
shock  of  the  whole  column  is  received  in  a  single  instant,  and  if 
no  escape  is  provided,  is  very  likely  to  burst  the  pipe.     The  inten- 
sity of  the  shock  of  water  when  stopped  is  made  the  means  of 
raising  a  portion  of  it  above  the  level  of  the  head.     The  instru- 
ment for  effecting  this  is  called  the  hydraulic  ram.     At  the  lower 
end  of  a  long  pipe,  P  (Fig.  148),  is  a  valve,  V,  opening  downward ; 

FIG.  148. 


144 


HYDRODYNAMICS. 


FIG.  149. 


near  it,  another  valve,  V,  opens  into  the  air-vessel,  A  ;  and  from 
this  ascends  the  pipe,  T,  in  which  the  water  is  to  be  raised.  As 
the  valve  V  lies  open  by  its  weight,  the  water  runs  out,  till  its 
momentum  at  length  shuts  it,  and  the  entire  column  is  suddenly 
stopped;  this  impulse  forces  the  water  into  the  air-vessel,  and 
thence,  by  the  compressed  air,  up  the  tube  T.  As  soon  as  the 
momentum  is  expended,  the  valve  V  drops,  and  the  process  is  re- 
peated. 

214.  Water- Wheels  with  a  Horizontal  Axis.— The 
•overshot  wheel  (Fig.  149)  is  con- 
structed with  buckets  on  the 
circumference,  which  receive 
the  water  just  after  passing  the 
highest  point,  and  empty  them- 
selves before  reaching  the  bot- 
tom. The  weight  of  the  water, 
as  it  is  all  on  one  side  of  a  ver- 
tical diameter,  causes  the  wheel 
to  revolve.  It  is  usually  made 
as  large  as  the  fall  will  allow, 
and  will  carry  machinery  with 
a  very  small  supply  of  water, 
if  the  fall  is  only  considerable. 
The  moment  of  each  bucket-full 
constantly  increases  from  a,  where  it  is  filled,  to  F,  where  its  act- 
ing distance  is  radius,  and  therefore  a  maximum.  From  F  down- 
ward the  moment  decreases,  both  by  loss 
of  water  and  diminution  of  acting  dis- 
tance, and  becomes  zero  at  L.  These 
wheels  deliver  from  70  to  80  per  cent,  of 
the  horse-power  of  the  fall  of  water  re- 
ceived upon  them. 

The  undershot  wheel  (Fig.  150)  is 
revolved  by  the  momentum  of  running 
water,  which  strikes  the  float-boards  on 
the  lower  side.  When  these  are  placed, 
as  in  the  figure,  perpendicular  to  the  circumference,  the  Avheel 
may  turn  either  way ;  this  is  the  construction  adopted  in  tide- 
mills.  When  the  wheel  is  required  to  turn  only  in  one  direction, 
an  advantage  is  gained  by  placing  the  float-boards  so  as  to  present 
an  acute  angle  toward  the  current,  by  which  means  the  water  acts 
partly  by  its  weight,  as  in  the  overshot  wheel.  The  undershot 
wheel  is  adapted  to  situations  where  the  supply  of  water  is  always 
abundant. 


FIG.  150. 


THE    TURBINE. 


145 


The  maximum  efficiency  of  these  wheels  is  obtained  when  the 
circumferential  velocity  is  one  half  the  velocity  of  the  water, 
and  is  about  30  per  cent,  of  the  theoretical  work  of  the  water 
used.  With  curved  float-boards  the  efficiency  may  reach  about 
60  per  cent. 

In  the  breast  wheel  (Fig.  151)  the  water  is  received  upon  the 
float-boards  at  about  the  height  of  the  axis,  and  acts  partly  by  its 
weight,  and  partly  by  its 

momentum.     The  planes  of  FIG.  151. 

the  float-boards  are  set  at 
right  angles  to  the  circum- 
ference of  the  wheel,  and 
are  brought  so  near  the  mill- 
course  that  the  water  is  held 
and  acts  by  its  weight,  as  in 
buckets.  The  efficiency  is 
about  40  to  50  per  cent. 

215.  The  Turbine.— 

This  very  efficient  water- 
wheel,  frequently  called  the 
French  turbine,  is  of  modern 

invention,  and  has  received  its  chief  improvements  in  this  coun- 
try. It  revolves  on  a  vertical  axis,  and  surrounds  the  bottom  of 
the  reservoir  from  which  it  receives  the  water.  The  lower  part 
of  the  reservoir  is  divided  into  a  large  number  of  sluices  by  curved 
partitions,  which  direct  the 
water  nearly  into  the  line  of  a 
tangent,  as  it  issues  upon  the 
wheel.  The  vanes  of  the  wheel 
are  curved  in  the  opposite  di- 
rection, so  as  to  receive  the 
force  of  the  issuing  streams  at 
right  angles.  The  horizontal 
section  (Fig.  152)  shows  the 
lower  part  of  the  reservoir  with 
its  curved  guides,  a,  a,  a,  and 
the  wheel  with  its  curved  vanes, 
v,  v,  v,  surrounding  the  reser- 
voir ;  D  is  the  central  tube, 
through  which  the  axis  of  the 

wheel  passes.  Fig.  153  is  a  vertical  section  of  the  turbine  ;  but 
it  does  not  present  the  guides  of  the  reservoir,  nor  the  vanes  of 
the  wheel.  C  G,  C  G,  is  the  outer  wall  of  the  reservoir  ;  D,  D, 
its  inner  wall  or  tube  ;  F,  F,  the  base,  curved  so  as  to  turn  the 


FIG.  152. 


1*6 


HYDRODYNAMICS. 


descending  water  gradually  into  a  horizontal  direction.      The 
outer  wall,  which  terminates  at  G,0,  is  connected  with  the  base 

FIG.  153. 


and  tube  by  the  guides  which  are  shown  at  a,  a,  in  Fig.  152. 
The  lower  rim  of  the  wheel,  H,  H,  is  connected  with  the  upper 
rim,  P,  P,  by  the  vanes  between  them,  v,  v  (Fig.  152),  and  to 
the  axis,  E,  E,  by  the  spokes  /,  /.  The  gate,  J,  /,•  is  a  thin 
•cylinder  which  is  raised  or  lowered  between  the  wheel  and  the 
sluices  of  the  reservoir.  The  bottom  of  the  axis  revolves  in  the 
socket  K,  and  the  top  connects  with  the  machinery.  As  the 
reservoir  cannot  be  supported  from  below,  it  is  suspended  by 
flanges  on  the  masonry  of  the  wheel-pit,  or  on  pillars  outside  of 
the  wheel.  To  prevent  confusion  in  the  figure,  the  supports  of 
the  reservoir  and  the  machinery  for  raising  the  gate  are  omitted. 
By  the  curved  base  and  guides  of  the  reservoir,  the  water  is  con- 
ducted in  a  spiral  course  to  the  wheel,  with  no  sudden  change  of 
direction,  and  thus  loses  very  little  of  its  force.  The  wheel  usually 
runs  below  the  level  of  the  water  in  the  wheel-pit,  as  represented 
in  the  figure,  L  L  being  the  surface  of  the  water.  The  reservoir 
is  sometimes  merely  the  extremity  of  a  large  tapering  tube  or 
supply  pipe,  bent  from  a  horizontal  to  a  vertical  direction.  In 
such  a  case,  the  tube  D  D,  in  which  the  axis  runs,  passes  through 
the  upper  side  of  the  supply  pipe.  The  figure  represents  only 
the  lower  part.  The  efficiency  is  about  80  per  cent,  though 
many  claim  a  much  higher  efficiency  than  this. 


BARKER'S    MILL. 


147 


FIG.  154. 


216.  Barker's  Mill. — This  machine  operates  on  the  prin- 
ciple of  unbalanced  hydrostatic  pressure.     It  consists  of  a  vertical 
hollow  cylinder,  A  B  (Fig.  154),  free  to  revolve  on  its  axis  M  N, 
and  having  a  horizontal  tube  connected 

with  it  at  the  bottom.  Near  each  end 
of  the  horizontal  tube,  at  P  and  P',  is 
an  orifice,  one  on  one  side,  and  one  on 
the  opposite.  The  cylinder,  being  kept 
full  of  water,  whirls  in  a  direction  op- 
posite to  that  of  the  discharging  streams 
from  P  and  P'.  This  is  owing  to  the 
fact  that  hydrostatic  pressure  .  is  re- 
moved from  the  apertures,  while  on  the 
interior  of  the  tube,  at  points  exactly 
opposite  to  them,  are  pressures  which 
are  now  unbalanced,  but  which  would 
be  counteracted  by  the  pressures  at  the 
apertures,  if  they  were  closed.  The 
tube  P  P'  may  revolve  either  in  the 
air,  or  beneath  the  surface  of  the  water. 
The  speed  of  rotation  is  increased  by  lengthening  the  tube  A  B.  , 

217.  Resistance  to  Motion  in  a  Liquid. — The  resistance 
which  a  body  encounters  in  moving  through  any  fluid  arises  from 
the  inertia  of  the  particles  of  the  fluid,  their  want  of  perfect  mo- 
bility among  each  other,  and  friction.     Only  the  first  of  these 
admits  of  theoretical  determination.     So  far  as  the  inertia  of  the 
fluid  is  concerned,  the  resistance  which  a  surface  meets  with  in 
moving  perpendicularly  through  it  varies  as  the  square  of  the 
velocity.     For  the  resistance  is  measured  by  the  momentum  im- 
parted by  the  moving  body  to  the  fluid.     And  this  momentum  (k) 
varies  as  the  product  of  the  quantity  of  fluid  set  in  motion  (m), 
and   its  velocity  (v) ;    or  k  oc    m  v.      But  it  is  obvious   that  the 
quantity  displaced  varies  as  the  velocity  of  the  body,  or  m  oc  v ; 
hence  k  cc  u2.     Therefore  the  resistance  varies  as  the  square  of 
the  velocity. 

This  proposition  is  found  to  hold  good  in  practice,  where  the 
velocity  is  small,  as  the  motions  of  boats  or  ships  in  water ;  but 
when  the  velocity  becomes  very  great,  as  that  of  a  cannon  ball, 
the  resistance  increases  in  a  much  higher  ratio  than  as  the  square 
of  the  velocity.  Since  action  and  reaction  are  equal,  it  makes  no 
difference,  in  the  foregoing  proposition,  whether  we  consider  the 
body  in  motion  and  the  fluid  at  rest,  or  the  fluid  in  motion  and 
striking  against  the  body  at  rest. 

Since  the  resistance  increases  so  rapidly,  there  is  a  wasteful 


148 


HYDRODYNAMICS. 


FIG.  155  (a). 


expenditure  of  force  in  trying  to  attain  great  velocities  in  naviga- 
tion. 

When  the  resistance  becomes  equal  to  the  moving  force,  the 
body  moves  uniformly,  and  is  said  to  be  in  a  state  of  dynamical 
equilibrium.  Thus,  a  body  falling  freely  through  the  air  by  grav- 
ity does  not  continue  to  be  accelerated  beyond  a  certain  limit,  but 
is  finally  brought,  by  the  resistance  of  the  air,  to  a  uniform  mo- 
tion. 

218.  Waves. — If  a  pebble  be  tossed  upon  still  water,  it  crowds 
aside  the  particles  beneath  it,  and  raises  them  above  the  level, 
forming  a  wave  around  it  in  the  shape  of  a  ring.     As  soon  as  this- 
ring  begins  to  descend,  it  elevates  above  the  level  another  portion 
around  itself,  and  thus  the  ring- wave  continues  to  spread  outward 
every  way  from  the  centre.     But  in  the  meantime  the  water  at  the 
centre,  as  it  rises  toward  the  level,  acquires  a  momentum  which 
lifts  it  above  that  level.     From  that  position  it  descends,  and  once 
more  passes  below  the  level,  thus  starting  a  new  *\vave  around  it, 
as  at  first,  only  of  less  height.     Hence,  we  see  as  the  result  of  the 
first  disturbance  a  series  of  concentric 

waves  continually  spreading  outward  and 
diminishing  in  height  at  greater  dis- 
tances, until  they  cease  to  be  visible. 
In  Fig.  155  (a)  are  represented  three  cir- 
cular waves  at  one  of  the  moments  of 
time  when  the  centre  is  lowest.  The 
shaded  parts  are  the  basins  or  troughs, 
and  the  light  parts,  c,  c,  c,  are  the  ridges 
or  crests.  Fig.  155  (b)  is  a  vertical  sec- 
tion along  the  line,  c,  c,  through  the 
centre  of  the  system,  corresponding  to  the  momentary  arrange- 
ment of  (a).  The  wave  centre  is  at  b,  and  the  crests  at  c,  c,  c.  A 
little  later,  when  either  crest  has  moved  FlQ  lgg  ,^ 

half  way  to  the  place  of  the  next  one,        -^^^fes^fes^isiTss^ss^ss 

both  figures  will  have  become  reversed :  lilL. 

the  centre  will  be  a  hillock,  the  troughs          c     c     c      c     c     c 
will  be   at  c,  c,  and   the  crests   at   the  middle   points   between 
them. 

219.  Molecular  Movements. — The  water  which  constitutes 
a  system  of  waves  does  not  advance  along  the  surface,  as  the  waves 
themselves  do  ;  for  a  floating  body  is  not  borne  along  by  them,  but 
alternately  rises  and  falls  as  the  waves  pass  under  it     Each  par- 
ticle of  water,  instead  of  advancing  with  the  wave,  describes  a  cir- 
cular path.     Within  short  distances  from  the  centre  of  disturbance 


PHASES.  149 

the  sizes  of  the  circles  described  by  the  surface  particles  are 
equal,  and  a  line  connecting  their  centres  would  be  parallel  to  the 
surface  of  the  still  water.  Upon  exciting  the  waves,  the  particles 
near  the  centre  are  first  disturbed  and  commence  their  journey 
around  the  circle.  The  neighboring  particles  are  then,  disturbed, 
and,  after  them,  their  neighbors.  All  the  particles  make  a  single 
revolution  in  the  same  time.  Particles  at  different  distances  from 
the  wave  centre  are,  at  a  given  instant,  in  different  positions  on 
their  respective  circles.  Fig;  156  is  supposed  to  be  a  vertical  sec- 
tion through  a  body  of  water  under  wave  disturbance.  The  wave 


1 h: ?- J i-  - J. u X-  —  u^ — j-irinrt 

«A        c<Ze^*oJfea'       JD'       o' 


is  progressing  in  the  direction  of  the  straight  arrow.  The  row  of 
circles  represents  the  paths  of  a  few  surface  particles.  The  direction 
of  rotation  of  the  particles  is  represented  by  the  curved  arrow.  The 
wave  is  to  be  regarded  as  already  started.  Particle  1  is  in  the  same 
position  as  it  would  be  were  there  no  disturbance,  and  is  just  about 
to  commence  its  rotation.  Particle  2  is  45°  behind  it  in  its  rota- 
tion, owing  to  its  not  having  been  agitated  as  soon  as  1.  Likewise 
3,  4,  5,  6,7,  and  8  are  90°,  135°,  180°,  225°,  270°,  and  315°  re- 
spectively behind  1.  No.  3  is  at  the  lowest  point  of  its  path,  and  is 
hence  at  the  centre  of  the  trough,  while  7  is  at  the  crest. 

Particles  below  the  surface,  as  far  as  the  wave  disturbance 
reaches,  perform  oscillations  synchronous  with  those  on  the  sur- 
face. Their  paths,  however,  are  smaller  circles,  or,  more  properly, 
ellipses  with  their  longer  axes  in  the  direction  of  the  wave  propa- 
gation. 

220.  Phases. — Wave  Length. — Whenever  two  particles, 
under  the  influence  of  wave  disturbance,  are  at  exactly  the  same 
points  in  their  respective  paths,  they  are  said  to  be  in  like  phase. 
Particles  1  and  1',  2  and  2'  (Fig.  156)  are  in  the  like  phase.  Both 
1  and  1'  are  just  about  to  commence  their  motions  from  their  po- 
sitions of  equilibrium.  Particles  which  are  at  diametrically  oppo- 
site points  of  their  respective  paths  are  said  to  be  in  opposite  phase. 
Particles  7  and  3  are  in  opposite  phases.  The  highest  points  of 


150  HYDRODYNAMICS. 

the  crests  of  two  waves  are  in  like  phase  ;  the  highest  point  of  the 
crest  and  the  lowest  point  of  the  trough  are  in  opposite  phase. 

The  length  of  the  straight  line  connecting  two  particles  in  like 
phase  is  termed  the  wave  length. 

221.  Water  Wave    Curve. — The  sectional  form  of  these 
waves  is  that  of  the  inverted  trochoid,  a  curve  described  by  a  point 
in  a  circle  as  it  rolls  on  a  straight  line.     The  curvature  of  the  crest 
is  always  greater  than  that  of  the  trough,  and  the  summit  may  pos- 
sibly be  a  sharp  ridge*,  in  which  case  the  section  of  the  trough  is  a 
cycloid,  the  describing  point  of  the  rolling  circle  being  on  the  cir- 
cumference ;  the  height  of  such  waves  is  to  their  length  as  the  di- 
ameter of  a  circle  to  the  circumference.     If  waves  are  ever  higher 
than  about  one-third  of  their  length,  the  summits  are  broken  into 
spray. 

222.  Velocity  of  Propagation. — Time  of  Oscillation. — 
During  the  time  that  a  wave  system  propagates  itself  through  the 
distance  of  one  wave  length,  a  particle,  which  happened  at  first  to 
be  at  a  crest,  rotates  through  the  lowest  part  of  the  trough  around 
to  the  crest  of  the  succeeding  wave  ;  during  this  time  it  has  made 
a  complete  revolution.     The  time  (T)  necessary  to  make  a  single 
revolution  is  termed  the  time  or  period  of  oscillation.     During  this 
time  the  wave  has  progressed  through  the  wave  length  (L).     If  we 
represent  the  velocity  of  the  propagation  by  c,  we  have 

c  =  ^,  L  =  c  Tand  T  =  L~. 

As  in  most  wave  motions  T  is  a  very  small  number,  it  is  more  con- 
venient to  employ  the  number  of  oscillations  made  in  one  second 
(n).  Evidently 

T  =. -,  c  =  n  L,  L  =  -  and  n  =  £. 
n  n  L 

Weber  says  that  the  velocities  of  propagation  in  fluids  of  differ- 
ent specific  gravity  (water  and  quicksilver)  are  very  nearly  equal. 
The  velocity  in  unconfined  waters,  however,  is  much  greater  than 
in  confined.  In  the  Atlantic  Ocean  the  velocity  is  about  13  meters, 
while  in  narrow  channels  it  is  about  0.75  meter. 

223.  Interference  of  Waves. — If  two  systems  of  waves  be 
simultaneously  excited,  the  waves  from  each  centre  will  be  propa- 
gated with  the  same  velocity  as  though  that  system  alone  were  act- 
ing.    An  affected  particle  will  describe  a  path,  which  is  the  result- 


REFLECTION    OF    WAVES. 


151 


ant  of  the  two  paths  which  it  would  describe  under  the  influence 
of  each  system  alone.  Thus  a  particle,  which  would  be  on  a  crest 
owing  to  one  system  and  on  a  crest  owing  to  the  other,  would  still 
be  on  a  crest,  but  of  greater  height  than  it  would  be  from  either 
one  alone.  The  reverse  would  be  the  result  of  two  combined 
troughs.  A  combination  of  crest  and  equal  trough  would  leave  the 
particle  at  rest.  The  results  of  this  coexistence  of  wave  motions 
are  termed  interferences. 


FIG.  151 


224.  Reflection  of  Waves. — If,  in  the  spreading  out  of  a 
wave,  a  particle  of  the  wave  medium,  in  endeavoring  to  describe  its 
circular  path,  strike  against 

a  fixed  barrier,  it  will,  be- 
cause of  its  elasticity,  re- 
bound and  move  circularly 
in  an  opposite  direction. 
The  ultimate  result  is  that 
a  new  set  of  waves,  moving 
in  an  opposite  direction,  is 
set  up,  and  they  are  said 
to  be  reflected  from  the  bar- 
rier. Let  waves  coming 
from  the  centre  A  (Fig. 
157)  meet  the  barrier  B  D. 
The  dotted  arcs  represent 
the  reflected  waves,  and 
they  appear  to  come  from 
a  centre,  C,  such  that  E  G 
—  A  E,  with  A  C  perpen- 
dicular to  B  D. 

It  must  be  borne  in  mind  that  the  planes  of  the  circular  paths 
are  perpendicular  to  the  wave  fronts,  i.e.,  they  embrace  the  radii 
from  A.  A  particle  at  B,  upon  striking  B  D,  not  only  reverses  the 
direction  of  its  rotation,  but  also  shifts  the  plane  of  its  path  so  as 
to  embrace  the  radius  G  C  from  C.  This  follows  naturally  from 
the  law  of  elasticity  (Art.  98). 

225.  Sea- Waves. — The  waves  raised  by  the  wind  rarely  ex- 
hibit the  precise   forms  above  described,  and  the  particles  rarely 
revolve  in  exact  circles,  partly  because  there  is  scarcely  ever  a  sys- 
tem of  waves  undisturbed   by  other  systems,  which  are  passing 
over  the  water   at  the   same  time,  and  partly  because  the  wind, 
which  was  the  original  cause  of  the  waves,  acts  continually  upon 
their  surfaces  to  distort  and  confuse  them. 


PART    III, 


CHAPTER    I. 

PROPERTIES    OF    GASES.— INSTRUMENTS    FOR   INVESTIGATION. 

226.  Gases  Distinguished  from   Liquids.— The  property 
of  mobility  of  particles,  which  belongs  to  all  fluids,  is  more  re- 
markable in  gases  than  in  liquids. 

While  gaseous  substances  are  compressed  with  ease,  they  are 
always  ready  to  expand  and  occupy  more  space.  This  property, 
called  dilatability,  scarcely  belongs  to  liquids  at  all. 

This  property  may  be  experimentally  illustrated  by  placing  a 
bag  only  partly  full  of  air  under  the  receiver  of  an  air  pump  and 
exhausting  the  air ;  the  external  pressure  having  been  removed, 
the  bag  will  seem  full  almost  to  bursting,  the  contained  air  hav- 
ing dilated  to  many  times  its  former  volume. 

Invert  a  flask  containing  air  into  a  beaker  of  colored  water, 
and  place  the  whole  under  the  receiver  of  an  air  pump.  As  the 
air  is  exhausted  the  contained  air  in  the  flask  will  expand  and, 
driving  the  water  out  of  the  neck  of  the  flask,  will  rise  in  bubbles 
to  the  surface.  Upon  admitting  air  again  to  the  receiver,  the 
water  will  be  forced  into  the  flask  to  take  the  place  of  the  escaped 
air,  and  will  rise  until  the  tension  of  the  contained  air,  together 
with  the  weight  of  the  water  column,  is  equal  to  that  of  the  air  in 
the  receiver. 

227.  Tension  of  Gases. — By  the  term  tension  just  used,  is 
meant  the  force  exerted  by  the  gas  at  each  instant  in  opposition 
to  any  compressing  or  restraining  force  ;  or,  in  other  words,  the 
force  of  expansion.     The  molecules  of  the  gas  are  supposed  to  be 
flying  through  space  with  great  velocity  in  straight  lines.     The 
combined  effect  of  the  impact  of  these  molecules  upon  the  walls 
of  the  containing  vessel  is  an  outward  pressure  which  is  opposed 
by  the  strength  of  the  material  of  the  vessel.     In  the  first  experi- 


OSMOSE    OF    GASES. 


153 


inent  given,  the  impact  of  the  molecules  of  the  air  in  the  room 
upon  the  outside  of  the  bag  counterbalanced  the  impact  of  the 
molecules  of  the  contained  air  upon  the  inside;  but  when  the 
external  air  was  removed,  there  was  no  counterbalancing  force, 
until  the  bag  expanded  so  much  that  the  strength  of  elasticity  of 
the  rubber  itself  equaled  the  resultant  of  the  impacts  within. 
This  thoory  of  tension  will  be  of  great  help  in  discussing  the  sub- 
ject of  expansion  by  heat. 

228.  Change  of  Condition. — Liquids,  and  even  solids,  may 
be  changed  into  the  gaseous  or  aeriform  condition  by  heating 
them  sufficiently.     By  being  cooled,  they  return  again  to  their 
former  state.     In  the  gaseous  form  they  are  called  vapors.    All 
substances  which  are  ordinarily  gases  can  be  so  far  cooled,  espe- 
cially under  great  pressure,  as  to  be  reduced  to  the  liquid  or  solid 
form. 

Those  which  can  only  be  thus  reduced  under  very  great  pres- 
sures, and  at  very  low  temperatures,  are  regarded  as  types  of  a 
theoretically  perfect  gas. 

229.  Diffusion  of  Gases.— If  two  flasks,  A  and  B,  be  con- 
nected by  a  tube,    as  in  Fig.   158,   and  the  FIG   159 
upper,  A,  be  filled  with  hydrogen,  or  illumi- 

natinggas,  and  the  lower  B  with  carbonic  diox-  p-IO  153 
ide,  after  a  time  some  of  the  lighter  gas  will  be 
found  in  B,  having  passed  down  through  the 
tube,  while  a  part  of  the  heavy  gas  in  B  will 
have  passed  upwards  to  A.  This  result  must 
follow  from  the  theory  of  molecular  motion 
given  before.  The  action  is  called  diffusion. 

230.  Osmose  of  Gases. — Cement  a  glass 
tube,  about  twenty-four  inches  long,  to  a  por- 
ous cell  B  (Fig.  159),  and  dip  the  lower  end 
of  the  tube  into  colored  liquid  A.     Now  fill  an 
inverted  bell  jar  with  hydrogen,  or  illuminat- 
ing  gas,    ami    place    it    over  B.     Either  will 
make  its  way  into  the  porous  cell  more  rapidly 
than  the  air  makes  its  way  out,  diffusion  in- 
wards being  more  rapid  than   diffusion   out- 
wards, and,  in  consequence,  some  air  will  be 
driven  out  through  the  glass  tube,  escaping  in 
bubbles  through  the  liquid  in  A.     Upon  re- 
moving the  bell  jar  the  gas  within  the  porous 
cell  will  pass  out  again  more  rapidly  than  air 
can  pass  in,   and  a  partial  vacuum   will  be 


154:  PNEUMATICS. 

formed,  causing  a  rise  of  the  colored  liquid  in  the  tube.  This 
mixing  of  gases  through  a  porous  cell,  or  a  thin  moistened  mem- 
brane, is  denominated  Osmose  of  Gases. 

Under  equal  pressures,  the  densities  of  gases  are  inversely  as 
the  squares  of  the  velocities,  with  which  equal  volumes  will  pass 
through  the  same  narrow  opening. 

231.  Weight  of  Gases. — Like  all  other  forms  of  matter, 
gases  have  weight.  Some  are  relatively  light,  some  heavy.  Take 
a  copper  globe,  and  hang  it  upon  one  scale  pan  of  a  delicate 
balance,  and  accurately  counterpoise  it.  Next  exhaust  the  air 
from  the  globe,  and  it  will  be  found  lighter  than  before ;  fill  with 
carbonic  dioxide,  and  it  will  weigh  much  more  than  at  first.  The 
heavier  gases  may  be  poured  from  one  vessel  to  another  like 
water ;  carbonic  dioxide  may  be  poured  from  a  beaker  upon  a 
burning  candle,  which  may  thus  be  extinguished. 

232. — Pressure  of  Gases.— As  a  consequence  of  the  weight 
of  gases  we  have  to  consider  the  pressure  exerted  by  them.  We 
shall  use  the  atmosphere  as  a  type  of  all  gases.  Across  the  open 
top  of  a  cylindric  receiver  stretch  a  sheet  of  rubber ;  upon  ex- 
hausting the  air  from  the  receiver,  the  rubber  will  be  pressed  in- 
wards by  the  external  air.  Substitute  for  the  sheet  of  rubber  a 
sheet  of  wetted  bladder,  which  allow  to  dry.  Upon  exhausting 
the  air  the  bladder  will  burst,  under  the  pressure  inwards,  with  a 
loud  report.  •  ' 

Exhaust  the  air  from  a  receiver,  into  which  projects  a  jet  tube- 
closed  with  a  stop-cock  ;  upon  submerging  the  outer  end  of  the 
jet  and  opening  the  stop-cock,  a  fountain  in  vacuo  will  be  pro- 
duced. 

Exhaust  the  air  from  two  closely  fitted  hemispheres,  called 
Magdeburg  hemispheres,  of  about  four  inches  diameter  ;  a  force 
of  over  175  Ibs.  will  be  required  to  separate  them. 

Having  a  cylinder  about  five  inches  in  diameter,  with  closely 
fitted  piston,  attach  a  weight  of  250  Ibs.  to  the  lower  side  of  the 
piston,  exhaust  the  air  from  the  cylinder  above  the  piston,  and 
the  weight  will  be  raised. 

The  pressure  of  the  air  upon  our  bodies  and  the  outward  pres- 
sure of  the  blood  against  the  walls  of  the  small  veins  and  capil- 
laries are  in  equilibrium.  Place  the  palm  of  the  hand  upon  the 
broad  opening  of  a  receiver,  called  a  hand  glass,  and  exhaust  the 
air  beneath ;  the  air  pressure  being  removed,  the  flesh  will  pro- 
trude into  the  receiver,  and  the  skin,  by  its  redness,  will  give 
evidence  of  the  engorgement  of  the  blood-vessels. 

233.  Buoyancy. — When  the  water  displaced  by  an  immersed! 


TORRICELLI'S    EXPERIMENT. 


155 


body  weighs  more  than  the  body  itself,  it  will  rise  to  the  surface 
arid  float ;  so  too  when  the  volume  of  air  displaced  by  a  body 
weighs  more  than  that  body,  it  will  rise  and  float  in  the  air.  At- 
tach the  gas  jet  to  a  clay  pipe  by  rubber  tubing,  and  blow  soap- 
bubbles  :  these  will  rise  rapidly  to  the  ceiling  of  the  room.  Blow 
a  small  bubble,  and  then  transfer  the  end  of  the  tube  to  the  gas 
jet  and  enlarge  the  bubble  till  its  specific  gravity  is  about  the 
same  as  that  of  the  air ;  it  will  now  float  about  the  room,  some- 
times rising,  sometimes  falling,  until  it  bursts. 

Large  balloons  have  ascended  to  a  height  of  seven  miles. 

If  a  large  and  a  small  body  are  in  equilibrium  on  the  two  arms 
of  a  balance,  and  the  whole  be  set  under  a  receiver,  and  the  air 
be  removed,  the  larger  body  will  preponderate,  showing  that  it  is. 
really  the  heaviest.  Their  apparent  equality  of  weight  when  in 
the  air  is  owing  to  its  buoyant  power,  which  diminishes  the 
apparent  weight  of  an  immersed  body  by  just  the  weight  of  the 
displaced  fluid.  Hence,  the  larger  the  body,  the  more  weight  it 


If  the  air  be  exhausted  from  a  tube,  four  or  five  feet  long  and 
two  inches  in  diameter,  containing  a  small  coin  and  a  feather,  it 
will  be  found,  upon  quickly  inverting  the  tube,  that  the  coin  and 
the  feather  will  fall  through  its  length  in  the  same  time,  both, 
having  the  same  velocity  ;  if  it  were  not 
for  the  obstruction  of  the  air,  all  bodies  FIG.  160. 

would  fall  to  the  earth  with  the  same 
velocity. 

234.  Torricelli's  Experiment. — 
A  glass  tube  A  B  (Fig.  160)  about  three 
feet  long,  and  hermetically  sealed  at 
one  end,  is  filled  with  mercury,  and 
then,  while  the  finger  is  held  tightly  on 
the  open  end,  it  is  inverted  in  a  cup  of 
mercury.  On  removing  the  finger  after 
the  end  of  the  tube  is  beneath  the  sur- 
face of  the  mercury,  the  column  sinks 
a  little  way  from  the  top,  and  there  re- 
mains. Its  height  is  found  to  be  nearly 
thirty  inches  above  the  level  of  mercury 
in  the  cup.  If  sufficient  care  is  taken  to 
expel  globules  of  air  from  the  liquid, 
the  space  above  the  column  in  the 
tube  is  as  perfect  a  vacuum  as  can  be 
obtained.  It  is  called  the  Torricellian 
vacuum,  from  Torricelli  of  Italy,  a 


156  PNEUMATICS. 

disciple  of  Galileo,  who,  by  this  experiment,  disproved  the  doctrine 
that  nature  abhors  a  vacuum,  and  fixed  the  limits  of  atmospheric 
pressure. 

235.  Pressure  of  Air  Measured. — The  column  is  sus- 
tained in  the  Torricellian  tube  by  the  pressure  of  air  on  the  sur- 
face of  mercury  in  the  vessel ;  for  the  level  of  a  fluid  surface 
cannot  be  preserved  unless  there  is  an  equal  pressure  on  every 
part.    Hence,  the  column  of  mercury  on  one  part,  and  the  column 
of  air  on  every  other  equal  part,  must  press  equally.    To  determine, 
therefore,  the  pressure  of  air,  we  have  only  to  weigh  the  column 
of  mercury,  and  measure  the  area  of  the  mouth  of  the  tube.     If 
this  is  carefully  done,  it  is  found  that  the  weight  of  mercury  is  about 
14.7  Ibs.  on  a  square  inch.     Therefore  the  atmosphere  presses  on 
the  earth  with  a  force  of  nearly  15  pounds  to  every  square  inch, 
or  more  than  2000  Ibs.  per  square  foot. 

The  specific  gravity  of  mercury  is  about  13. 6;  and  therefore 
the  height  of  a  column  of  water  in  a  Torricellian  tube  should  be 
13.6  times  greater  than  that  of  mercury,  that  is,  about  34  feet. 
Experiment  shows  this  to  be  true.  And  it  was  this  significant 
fact,  that  equal  weights  of  water  and  mercury  are  sustained  in 
these  circumstances,  which  led  Torricelli  to  attribute  the  effect  to 
a  common  force,  namely,  the  pressure  of  the  air. 

236.  Pascal's  Experiment. — As  soon  as  Torricelli's  discov- 
ery was  known,  Pascal  of  France  proposed  to  test  the  correct- 
ness of  his  conclusion,  by  carrying  the  apparatus  to  the  top  of  a 
mountain,  in  order  to  see  if  less  air  above  the  instrument  sustained 
the  mercury  at  a  less  height.     This  was  found  to  be  true;  the 
column  gradually  fell,  as  greater  heights  were  attained.     The  ex- 
periment of  Pascal  also  determined  the  relative  density  of  mer- 
cury and  air.     For  the  mercury  falls  one-tenth  of  an  inch  in 
ascending  87.2  feet;  therefore  the  weight  of  the  one-tenth  of  an 
inch  of  mercury  was  balanced  by  the  weight  of  the  87.2  feet  of 
air.    Therefore  the  specific  gravities  of  mercury  and  air  (being  in- 
versely as  the  heights  of  columns  in  equilibrium)  are  as  (87.2  x  12 
x  10  =)  10464  :  1.     In  the  same  way  it  is  ascertained  that  water 

is  770  times  as  dense  as  air.  These  results  can  of  course  be  con- 
firmed by  directly  weighing  the  several  fluids,  which  could  not  be 
done  before  the  invention  of  the  air-pump. 

That  it  is  the  atmospheric  pressure  which  sustains  the  column 
of  mercury  may  be  shown  thus :  Place  the  Torricellian  tube  and 
cistern  under  a  receiver,  made  for  the  purpose,  and  exhaust  the 
air ;  the  mercury  will  fall  lower  and  lower  at  each  stroke  of  the 
pump,  until,  if  the  pump  be  in  good  working  order,  the  column 
will  be  nearly  at  the  level  of  the  mercury  in  the  cistern. 


MARIOTTE'S    LAW. 


157 


FIG.  161. 


237.  Mariotte's  Law.— 

At  a  given  temperature,  the  volume  of  air  is  inversely  as  the 
compressing  force. 

An  instrument  constructed  for  showing  this  is  called  Mariotte's 
tube.  The  end  B  (Fig.  161)  is  sealed,  and  A  open.  Pour  in  small 
quantities  of  mercury,  inclining  the  tube  so  as  to  let  air  in  or  out, 
till  both  branches  are  filled  to  the  zero  point.  The  air  in  the 
short  branch  now  has  the  same  tension  as  the  external  air,  since 
they  just  balance  each  other.  If  mercury  be  poured  in  till  the 
column  in  the  short  tube  rises  to  (7,  the  inclosed  air  is  reduced  to 
one-half  of  its  original  volume,  and  the  column  A  in  the  long  branch 
is  found  to  be  29  or  30  inches  above  the  level  of  C,  according  to 
the  barometer  at  the  time.  Thus,  two  atmospheres,  one  of  mer- 
cury, the  other  of  air  above  it,  have  compressed  the  inclosed  air 
into  one-half  its  volume.  If  the 
tube  is  of  sufficient  length,  let 
mercury  be  poured  in  again,  till 
the  air  is  compressed  to  one- 
third  of  its  original  space;  the 
long  column,  measured  from 
the  level  of  the  mercury  in  the 
short  one,  is  now  twice  as  high 
as  before;  that  is,  three  atmos- 
pheres, two  of  mercury  and  one 
of  air,  have  reduced  the  same 
quantity  of  air  to  one-third  of  its 
first  volume.  This  law  has  been 
found  to  hold  good  in  regard  to 
atmospheric  air  up  to  a  pressure 
of  nearly  thirty  atmospheres. 

On  the  other  hand,  if  the 
pressure  on  a  given  mass  of  air 
is  diminished,  its  volume  is 
found  to  increase  according  to 
the  same  law.  When  the  pres- 
sure is  half  an  atmosphere,  the 
volume  is  doubled;  when  one- 
third  of  an  atmosphere,  the 
volume  is  three  times  as  great, 
&c. 

This  may  be  shown  by  filling 
a  tube  A,  closed  at  one  end,  nearly  full  of  mercury,  and  inverting 
it  in  a  tube-like  cistern  B,  as  in  Fig.  162.  Suppose  that  the  tube, 
of  uniform  bore,  contains  an  inch  of  air  when  the  mercury  is  at 


158  PNEUMATICS. 

the  same  level  within  and  without ;  upon  raising  the  tube  until 
the  column  within  is  about  fifteen  inches  high  the  air  will  be 
found  to  occupy  two  inches  of  space.  At  the  beginning  of  the 
experiment  the  mercury  being  at  the  same  level  in  both  tubes,  the 
pressure  upon  the  contained  air  was  due  to  the  atmosphere,  and 
was  about  14.7  Ibs.  per  square  inch.  At  the  close  the  pressure 
was  less  by  the  force  necessary  to  sustain  the  fifteen  inches  of 
mercury,  leaving  only  a  pressure  of  about  one-half  an  atmosphere. 
Experiments  in  this  case  will  not  be  satisfactory  unless  precau- 
tions be  taken  to  remove  the  air  bubbles  which  adhere  to  the  glass 
tube.  Since  the  tension  of  the  inclosed  air  always  balances  the 
compressing  force,  and  since  the  density  is  inversely  as  the 
volume,  it  follows  from  Mariotte's  Law  that,  when  the  tempera- 
ture is  the  same, 

The  tension  of  air  varies  as  the  compressing  force ;  and  TJie 
tension  of  air  varies  as  its  density. 

This  la\v  is,  however,  not  strictly  true  of  gases,  except  when 
far  removed  from  the  critical  point;  when  near  the  point  of  con- 
densation the  departure  from  the  law  is  most  marked.  Even  in 
the  ordinary  state  the  law  is  not  strictly  followed  by  many  gases. 

238.  Dalton's  Law.— 

At  a  given  temperature  the  tension  of  a  mixture  of  gases  is 
equal  to  the  sum  of  the  tensions  of  the  gases  taken  separately. 

In  this  law  a  mixture  is  spoken  of  and  not  a  chemical  combi- 
nation. Each  gas  diffuses  and  is  found  equally  distributed 
throughout  the  containing  vessel  just  as  though  no  other  gas  were 
present,  differing  in  this  respect  from  mechanical  mixtures  of 
liquids,  such  as  oil  and  water,  in  which  the  components  of  the 
mixture  arrange  themselves  according  to  their  specific  gravities. 
If  a  vessel  A  contains  a  cubic  foot  of  nitrogen  at  a  tension  often 
Ibs.  per  square  inch,  and  a  perfect  vacuum  B  of  the  same  capacity 
be  connected  with  this,  and  all  the  gas  be  transferred  to  the  new 
vessel  B,  the  tension  in  the  latter  case  will  be  the  same  as  in  the 
former,  ten  Ibs.  Now,  if  a  cubic  foot  of  oxygen  at  a  tension  of 
ten  Ibs.  be  transferred  to  the  vessel  B  also,  it  will  exert  a  pressure 
of  ten  Ibs.  just  as  though  the  nitrogen  were  not  present,  giving  a 
total  pressure  of  twenty  Ibs.  for  the  mixture.  These  two  illustra- 
tions have  been  given  to  prevent  any  misapprehension  which  may 
arise  from  the  following  frequently  repeated  and  very  concise 
wording  of  the  law  :  "  Every  gas  acts  as  a  vacuum  with  respect  to- 
every  other." 

239.  Laws  of  Mixture  of  Gases  and  Liquids.— Water 

and  many  other  liquids  will  contain  gases  in  solution  ;  but  under 
the  same  conditions  of  temperature  and  pressure,  a  given  liquid 


MIXTURE    OF    GASES    AND    LIQUIDS.  159 

does  not  absorb  equal  quantities  of  different  gases.  For  example, 
at  the  mean  temperature  and  pressure  water  dissolves  about  .025 
of  its  volume  of  nitrogen,  .046  of  its  volume  of  oxygen,  its  own 
volume  of  carbonic  dioxide,  and  430  times  its  volume  of  am- 
monia. Mercury,  on  the  other  hand,  does  not  dissolve  any  of 
the  gases. 

Experiment  has  determined  the  three  following  laws  of  the 
mixture  of  liquids  and  gases. 

1st.  For  the  same  gas,  the  same  liquid,  at  a  constant  tempera- 
ture, the  weight  of  gas  absorbed  is  proportional  to  the  pressure. 
From  this  it  follows  that  the  volume  dissolved  is  constant,  what- 
ever may  be  the  changes  in  pressure,  or,  what  is  the  same  thing, 
the  density  of  the  gas  absorbed  bears  a  constant  ratio  to  that  of 
the  gas  not  absorbed. 

2d.  The  quantity  of  gas  absorbed  increases  as  the  temperature 
decreases. 

3d.  The  quantity  of  a  gas  which  a  given  liquid  will  dissolve  is 
independent  of  the  kinds  and  quantities  of  other  gases  which  may 
already  be  held  in  solution. 

If  instead  of  a  single  gas  in  contact  with  the  liquid,  a  mix- 
cure  of  several  gases  be  used,  each  of  these  will  be  dissolved  in  the 
quantity  due  to  its  proportional  part  of  the  total  pressure.  For 
example,  since  oxygen  forms  only  about  one-fifth  the  volume  of 
the  air,  water  under  ordinary  conditions  absorbs  the  same  quantity 
of  oxygen,  as  if  the  atmosphere  were  wholly  of  oxygen  under 

.  14.7,, 

a  pressure  of  — —  Ibs. 
5 

The  first  law  may  be  experimentally  illustrated  by  opening  a 
bottle  of  common  soda  water.  As  soon  as  the  cork  is  loosened 
it  is  driven  out  by  the  tension  of  the  confined  carbonic  dioxide 
above  the  liquid,  and  the  pressure  being  reduced  by  the  escape  of 
the  free  gas,  the  absorbed  gas  is  at  once  given  off  in  bubbles, 
the  escape  of  which  produces  foaming. 

If  after  all  the  gas  has  seemingly  escaped,  a  portion  of  the 
liquid  be  poured  into  a  beaker  and  placed  under  the  receiver  of 
the  air  pump,  a  fresh  discharge  of  bubbles  will  follow  the  first 
stroke  of  the  pump,  consequent  upon  the  still  further  reduction 
of  pressure. 

A  portion  poured  into  a  flask  and  heated  will  serve  to  illustrate 
the  second  law,  the  rise  in  temperature  causing  a  constant  rise  of 
bubbles  of  gas. 

240.  The  Barometer. — When  the  Torricellian  tube  and 
basin  are  mounted  in  a  case,  and  furnished  with  a  graduated  scale, 
the  instrument  is  called  a  barometer.  The  scale  is  divided  into 


160  PNEUMATICS. 

millimeters  or  inches  and  tenths,  and  usually  extends  for  a  space 
more  than  sufficient  to  include  all  the  natural  variations  in 
the  weight  of  the  atmosphere.  By  attaching  a  vernier  to  the 
scale,  as  is  commonly  done  in  meteorological  observations,  the 
reading  may  be  carried  to  the  fractional  parts  of  the  scale. 
By  observing  the  barometer  from  day  to  day,  and  from  hour 
to  hour,  it  is  found  that  the  atmospheric  pressure  is  constantly 
fluctuating. 

As  the  meteorological  changes  of  the  barometer  are  all  com- 
prehended within  a  range  of  two  or  three  inches,  much  labor  has 
been  expended  in  devising  methods  for  magnifying  the  motions 
of  the  mercurial  column,  so  that  more  delicate  changes  of  atmos- 
pheric pressure  might  be  noted.  The  inclined  tube  and  the  wheel 
barometer  are  intended  for  this  purpose.  A  description  of  these 
contrivances,  however,  is  unnecessary,  as  they  are  all  found  to  be 
inferior  in  accuracy  to  the  simple  tube  and  basin. 

241.  Corrections  for  the  Barometer. — 

1.  For  change  of  level  in  the  basin. — The  numbers  on  the 
barometer  scale  are  measured  from  a  certain  zero  point,  which  is 
assumed  to  be  the  level  of  the  mercury  in  the  basin.     If  now  the 
column  falls,  it  raises  the  surface  in  the  basin ;  and  if  it  rises,  it 
lowers  it.    If  the  basin  is  broad,  the  change  of  level  is  small,  but 
it  always  requires  a  correction.     To  avoid  this  source  of  error,  the 
bottom  of  the  basin  is  made  of  flexible  leather,  with  a  screw  under- 
neath it,  by  which  the  mercury  may  be  raised  or  lowered,  till  its 
surface  touches  an  index  that  marks  the  zero  point.     This  adjust- 
ment should  always  be  made  before  reading  the  barometer. 

2.  For  capillarity. — In  a  glass  tube  mercury  is  depressed  by 
capillary  action  (Art.   200).     The  amount  of  depression  is  less  as 
the  tube  is  larger.     This  error  is  to  be  corrected  by  the  manufac- 
turer, the  scale  being  put  below  the  true  height  by  a  quantity  equal 
to  the  depression. 

There  is  a  slight  variation  in  this  capillary  error,  arising  from 
the  fact  that  the  rounded  summit  of  the  column,  called  the  menis- 
cus, is  more  convex  when  ascending  than  when  descending.  To 
render  the  meniscus  constant  in  its  form,  the  barometer  should  be 
jarred  before  each  reading. 

3.  For  temperature. — As  mercury  is  expanded  by  heat  and 
contracted  by  cold,  a  given  atmospheric  pressure  will  raise  the 
column  too  high,  or  not  high  enough,  according  to  the  tempera- 
ture of  the  mercury.    A  thermometer  is  therefore  attached  to  the 
barometer,  to  show  the  temperature  of  the  instrument.    By  a  table 
of  corrections,  each  reading  is  reduced  to  the  height  the  mercury 
would  have  if  its  temperature  was  0°  C. 


THE     ANEROID     BAROMETER. 


161 


FIG.  163. 


FIG.  164. 


4.  For  altitude  of  station. — Before  comparing  the  observations 
of  different  places,  a  correction  must  be  made  for  altitude  of  sta- 
tion, because  the  column  is  shorter  according  as  the  place  is  higher 
above  the  sea  level. 

242.  The  Aneroid  Barometer.— This  is  a  small  and  port- 
able instrument,  in  appearance  a  little  like  a  large  chronometer. 
The  essential  part  of  this  barometer  is  a  flat  cylindrical  metallic 
box,  shown  in  section  at  A   (Fig.   163), 

whose  upper  surface  is  corrugated,  so  as 
to  be  yielding. 

The  box  being  partly  exhausted  of  air, 
the  external  pressure  causes  the  top  to 
sink  in  to  a  certain  extent ;  if  the  pres- 
sure increases,  the  surface  descends  a 
little  more  ;  if  it  diminishes,  a  little  less. 

These  small  movements  are  communicated  to  a  lever  b  (Fig. 
163  )  whose  end  c  moves  over  a  scale  of  inches  on  the  case  (Fig. 
164).  To  insure  contact  of  the  pin  o 
(Fig.  163),  and  also  to  secure  a  constant 
resistance  to  the  motion  of  the  lever  b  a 
springe?  is  attached  to -the  lever  b  near 
the  fulcrum,  which  is  pressed  upon  by  the 
screw  5,  whose  graduated  head  H  is  of 
the  same  diameter  as  the  box.  The  end 
of  this  spring  e  (Fig.  164)  must  be 
brought  to  coincide  with  the  end  of  the 
lever,  by  turning  the  screw-head  H.  The 
reading  of  inches  and  tenths  is  taken 
from  the  scale  of  inches,  the  hundredths  and  thousandths  being 
given  by  the  screw-head.  This  is  one  of  the  most  simple  of  all 
the  aneroids  in  construction.  A  table  of  corrections  for  tempera- 
ture, and  reductions  to  the  standard  mercurial  barometer  is  entered 
upon  the  cover. 

243.  Pressure  and   Latitude. — The  mean  pressure  of  the 
atmosphere  at  the  level  of  the  sea  is  761  mm.  or  30  inches.     But 
it  is  not  the  same  at  all  latitudes.     From  the  equator  either  north- 
ward or  southward,  the  mean  pressure  increases  to  about  latitude 
30°,  by  4  or  5  millimeters,  and  thence  decreases  to  about  65°, 
where  the  pressure  is  less  than  at  the  equator,  and  beyond  that  it 
slightly  increases.     This  distribution  of  pressures  in  zones  is  due 
to  the  great  atmospheric  currents,  caused  by  heat  in  connection 
with  the  earth's  rotation  on  its  axis. 

The  amount  of  variation  in  barometric  pressure  is  very  unequal 
in  different  latitudes  ;  and  in  general,  the  higher  the  latitude,  the 
11 


162  PNEUMATICS. 

greater  the  variation.  Within  the  tropics  the  extreme  range 
scarcely  ever  exceeds  one-fourth  of  an  inch,  while  at  latitude  40° 
it  is  more  than  two  inches,  and  in  higher  latitudes  even  reaches 
three  inches. 

244.  Diurnal  Variation. — If  a  long  series  of  barometric  ob- 
servations be  made,  and  the  mean  obtained  for  each  hour  of  the 
day,  the  changes  caused  by  weather  become  eliminated,  and  the 
diurnal  oscillation  reveals  itself.  It  is  found  that  the  pressure 
reaches  a  maximum  and  a  minimum  twice  in  24  hours.  The 
times  of  greatest  pressure  are  from  9  to  11,  and  of  least  pressure 
from  3  to  5,  both  A.  M.  and  P.  M.  In  tropical  climates  this  varia- 
tion is  very  regular,  though  small ;  but  in  the  temperate  zones  the 
irregular  fluctuations  of  weather  conceal  it  in  a  great  degree. 

This  daily  fluctuation  of  the  barometer  is  caused  by  the 
changes  which  take  place  from  hour  to  hour  of  the  day  in  the 
temperature,  and  by  the  varying  quantity  of  vapor  in  the  atmos- 
phere. 

The  surface  of  the  globe  is  always  divided  into  a  day  and  night 
hemisphere,  separated  by  a  great  circle  which  revolves  with  the 
sun  from  east  to  west  in  twenty-four  hours.  The  hemisphere  ex- 
posed to  the  sun  'is  warm,  the  other  is  cold.  The  time  of  greatest 
heat  is  not  at  noon,  when  the  sun  is  in  the  meridian,  but  about 
two  or  three  hours  after  ;  the  period  of  greatest  cold  occurs  about 
four  in  the  morning.  As  the  hemisphere  under  the  sun's  rays 
becomes  heated,  the  air,  expanding  upwards  and  outwards,  flows 
over  upon  the  other  hemisphere  where  the  air  is  colder  and  denser. 
There  thus  revolves  round  the  globe  from  day  to  day  a  wave 
of  heated  air,  from  the  crest  of  which  air  constantly  tends  to  flow 
towards  the  meridian  of  greatest  cold  on  the  opposite  side  of  the 
the  globe. 

The  barometer  is  influenced  to  a  large  extent  by  the  elastic 
force  of  the  vapor  of  water  invisibly  suspended  in  the  atmosphere, 
in  the  same  way  as  it  is  influenced  by  the  dry  air  (oxygen  and 
nitrogen).  But  the  vapor  of  water  also  exerts  a  pressure  on  the 
barometer  in  another  way.  Vapor  tends  to  diffuse  itself  equally 
through  the  air  ;  but  as  the  particles  of  air  offer  an  obstruction  to 
the  watery  particles,  about  9  or  10  A.  M.,  when  evaporation  is 
most  rapid,  the  vapor  is  accumulated  or  pent  up  in  the  lower 
stratum  of  the  atmosphere,  and  being  impeded  in  its  ascent  its 
elastic  force  is  increased  by  the  reaction,  and  the  barometer  con- 
sequently rises.  When  the  air  falls  below  the  temperature  of  the 
dew-point  part  of  its  moisture  is  deposited  in  dew,  and  since  some 
time  must  elapse  before  the  vapor  of  the  upper  strata  can  diffuse 
itself  downwards  to  supply  the  deficiency,  the  barometer  falls, — 


HEIGHTS    MEASURED    BY    THE    BAROMETER.    163 

most  markedly  at   10   p.    H.,    when  the   deposition   of  dew  is 
greatest. 

245.  The  Barometer  and  the  Weather. — The  changes 
in  the  height  of  the  barometer  column  depend  directly  on  nothing 
else  than  the  atmospheric  pressure.   But  these  changes  of  pressure 
are  due  to  several  causes,  such  as  wind  and  changes  of  temperature 
and  moisture. 

The  practice  formerly  prevailed  of  engraving  at  different  points 
of  the  barometer  scale  several  words  expressive  of  states  of  weather, 
"fair,  rain,  frost,  wind,"  &c.  But  such  indications  are  worthless, 
being  as  often  false  as  true  ;  this  is  evident  from  the  fact  that  the 
height  of  the  column  would  be  changed  from  one  kind  of  weather 
to  another  by  simply  carrying  the  instrument  to  a  higher  or  lower 
station. 

Xo  general  system  of  rules  can  be  given  for  anticipating  changes 
of  weather  by  the  barometer,  which  would  be  applicable  in  differ- 
ent countries.  Eules  found  in  English  books  are  of  very  little 
value  in  America. 

Severe  and  extensive  storms  are  almost  always  accompanied 
by  a  fall  of  the  barometer  while  passing,  and  succeeded  by  a  rise 
of  the  barometer. 

246.  Heights  Measured  by  the  Barometer. — Since  mer- 
cury is  10464  times  as  heavy  as  air  (Art.  236),  if  the  barometer  is 
carried  up  until  the  mercury  falls  one  inch,  it  might  be  inferred 
that  the  ascent  is  10464  inches,  or  872  feet.     This  would  be  the 
case  if  the  density  were  the  same  at  all  altitudes.    But,  on  account 
of  diminished  pressure,  the  air  is  more  and  more  expanded  at 
greater  heights.    Besides  this,  the  height  due  to  a  given  fall  of  the 
mercury  varies  for  many  reasons,  such  as  the  temperature  of  the 
air,  the  temperature  of  the  mercury,  the  elevation  of  the  stations, 
and  their  latitude.     Hence,  the  measurement  of  heights  by  the 
barometer  is  somewhat  troublesome,  and  not  always  to  be  relied 
on.      Formulae  and  tables  for  this  purpose  are  to  be  found  in 
practical  works  on  physics. 

247.  The  Gauge  of  the  Air-Pump.—  The  Torricellian  tube 
is  employed  in  different  ways  as  a  gauge  for  the  air-pump,  to  indi- 
cate the  degree  of  exhaustion.     In  Fig.  166  the  gauge  G  is  a  tube 
about  33  inches  long,  both  ends  of  which  are  open,  the  lower  im- 
mersed in  a  cup  of  mercury,  and  the  upper  communicating  with 
the  interior  of  the  receiver.     As  the  exhaustion  proceeds,  the 
pressure  is  diminished  within  the  tube,  and  the  external  air  raises 
the  mercury  in  it.     A  perfect  vacuum  would  be  indicated  by  a 
height  of  mercury  equal  to  that  of  the  barometer  at  the  time. 


164 


PNEUMATICS. 


Another  kind  of  gauge  is  a  barometer  already  filled,  the  basin 
of  which  is  open  to  the  receiver.  As  the  tension  of  air  in  the 
receiver  is  diminished,  the  column  descends,  and  would  stand  at  the 
same  level  in  both  tube  and  basin,  if  the  vacuum  were  perfect. 

A  modified  form  of  the  last,  called  the  siphon  gauge,  „ 
is  the  best  for  measuring  the  rarity  of  the  air  in  the 
•  receiver  when  the  vacuum  is  nearly  perfect.  Its  con- 
struction is  shown  by  Fig.  165.  The  top  of  the  column, 
A,  is  only  5  or  6  inches  above  the  level  of  B  in  the  other 
branch  of  the  recurved  tube.  As  the  air  is  withdrawn 
from  the  open  end  C,  the  tension  at  length  becomes  too 
feeble  to  sustain  the  column  ;  it  then  begins  to  descend, 
and  the  mercury  in  the  two  branches  approaches  a  com- 
mon level. 


CHAPTER    II. 

INSTRUMENTS  WHOSE  OPERATION  DEPENDS  ON  THE  PROPER- 
TIES   OF    AIR. 

248.  The   Air-Pump. — This  is  an  instrument  by  which 
nearly  all  the  air  can  be  removed  from  a  vessel  or  receiver.    It  has 
a  variety  of  forms,  one  of  which  is  shown  in  Fig.  166.     In  the 
FIG.  166. 


( 
THE    AIR-PUMP.  105 

barrel  B  an  air-tight  piston  is  alternately  raised  and  depressed  by 
the  lever,  the  piston-rod  being  kept  vertical  by  means  of  a  guide. 
The  pipe  P  connects  the  bottom  of  the  barrel  with  the  brass  plate 
L,  on  which  rests  the  receiver  R.  The  surface  of  the  plate  and 
the  edge  of  the  receiver  are  both  ground  to  a  plane.  G  is  the 
gauge  which  indicates  the  degree  of  exhaustion.  There  are  three 
valves,  the  first  at  the  bottom  of  the  barrel,  the  second  in  the 
piston,  and  the  third  at  the  top  of  the  barrel.  These  all  open 
upward,  allowing  the  air  to  pass  out,  but  preventing  its  return. 

249.  Operation. — When  the  piston  is  depressed,  the  air  below 
it,  by  its  increased  tension,  presses  down  the  first  valve,  and  opens 
the  second,  and  escapes  into  the  upper  part  of  the  barrel.     When 
the  piston  is  raised,  the  air  above  it- cannot  return,  but  is  pressed 
through  the  third  valve  into  the  open  air ;  while  the  air  in  the  re- 
ceiver and  pipe,  by  its  tension,  opens  the  first  valve,  and  diffuses 
itself  equally  through  the  receiver  and  barrel.     Another  descent 
and  ascent  only  repeat  the  same  process  ; 

and  thus,  by  a  succession  of  strokes,  the  FlG-  16 '  • 

air  is  nearly  all  removed. 

The  exhaustion  can  be  made  more 
complete  if  the  first  and  second  valves 
are  opened  by  the  action  of  the  piston 
and  rod,  rather  than  by  the  tension  of 
the  air.  This  method  is  illustrated  by 
Fig.  167,  a  section  of  the  barrel  and  pis- 
ton. The  first  and  second  valves,  as 
shown  in  the  figure,  are  conical  or  pup- 
pet valves,  fitting  into  conical  sockets. 
The  first' has  a  long  stem  attached,  which 
passes  through  the  piston  air-tight,  and 
is  pulled  up  by  it  a  little  way,  till  it  is 
arrested  by  striking  the  top  of  the  barrel. 
The  second  valve  is  a  conical  frustum  on 
the  end  of  the  piston-rod.  When  the 
rod  is  raised,  it  shuts  the  valve  before 
moving  the  piston  ;  when  it  begins  to  descend,  it  opens  the  valve 
again  before  giving  motion  to  the  piston.  The  first  valve  is  shut 
by  a  lever,  which  the  piston  strikes  at  the  moment  of  its  reaching 
the  top.  The  oil  which  is  likely  to  be  pressed  through  the  third 
valve  is  drained  off  by  the  pipe  (on  the  right  in  both  figures)  into 
a  cup  below  the  pump. 

250.  Rate  of  Exhaustion.— The  quantity  removed,  by  suc- 
cessive strokes,  and  also  the  quantity  remaining  in  the  receiver, 
diminishes  in  the  same  geometrical  ratio.     For,  of  the  air  occupy- 


166 


PNEUMATICS. 


FIG.  168. 


ing  the  barrel  and  receiver,  a  barrel-full  is  removed  at  each  stroke, 
and  a  receiver-full  is  left.  If,  for  example,  the  receiver  is  three 
times  as  large  as  the  barrel,  the  air  occupies  four  parts  before  the 
descent  of  the  piston ;  and  by  the  first  stroke  one-fourth  is  re- 
moved, and  three-fourths  are.,  left.  By  the  next  stroke,  three- 
fourths  as  much  will  be  removed  as  before  (£  of  £ ,  instead  of  £  of 
the  whole),  and  so  on  continually.  The  quantity  left  obviously 
diminishes  also  in  the  same  ratio  of  three-fourths.  In  general,  if 
b  expresses  the  capacity  of  the  barrel,  and  r  that  of  the  receiver 

and  connecting-pipe,  the  ratio  of  each  descending  series  is-y . 

With  a  given  barrel,  the  rate  of  exhaustion  is  obviously  more  rapid 
as  the  receiver  is  smaller.     If  the  two  were  equal,  ten  strokes  would 
rarefy  the  air  more  than  a  thousand  times.     For  (£)10  =  yAr- 
As  a  term  of  this  series  can  never  reach  zero,  a  complete  ex- 
haustion can  never  be  effected  by  the  air-pump ; 
but  in  the  best  condition  of  a  well-made  pump, 
it  is  not  easy  to  discover  by  the  gauge  that  the 
vacuum  is  not  perfect. 

251.  Sprengel's  Pump.— This  apparatus  is 
too  slow  in  its  action  for  ordinary  lecture  illustra- 
tion, but  gives  a  much  better  vacuum  than  any 
piston  pump.  The  length  a  b  (Fig.  168)  must  be 
more  than  30  inches,  and  the  diameter 
of  the  tube  should  be  quite  small,  about 
•jV  inch.  Mercury  from  the  funnel  F 
falls  down  the  tube  a  b  in  drops,  which 
carry  air  before  them  from  the  receiver, 
which  is  connected  with  the  exhaust 
branch  B  by  suitable  tubing. 

252.  The  Air  Condenser.— While 
the  air-pump  shows  the  tendency  of  air 
to  dilate  indefinitely,  as  the  compressing 
force  is  removed,  another  useful  instru- 
ment, the  condenser,  exhibits  the  in- 
definite compressibility  of  air.  Like 
the  pump,  it  consists  of  a  barrel  and 
piston,  but  its  valves,  one  in  the  piston 
and  one  at  the  bottom  of  the  barrel, 
open  downward.  Fig.  169  shows  the 
exterior  of  the  instrument.  If  it  be  screwed  upon  the  top  of  a 
strong  receiver  (Fig.  170),  with  a  stop-cock  connecting  them,  air 
may  be  forced  in,  and  then  secured  by  shutting  the  stop-cock. 


FIG.  169. 


AIR    CONDENSER. 


167 


When  the  piston  is  depressed,  its  own  valve  is  shut  by  the  in- 
creased tension  of  the  air  beneath  it,  and  the  lower 
one  opened  by  the  same  force.  When  the  piston 
is  raised,  the  lower  valve  is  kept  shut  by  the  con- 
densed air  in  the  receiver,  and  that  of  the  piston 
is  opened  by  the  weight  of  the  outer  air,  which 
thus  gets  admission  below  the  piston. 

The  quantity  of  air  in  the  receiver  increases  at 
each  stroke  in  an  arithmetical  ratio,  because  the 
same  quantity,  a  barrel-full  of  common  air,  is  added 
every  time  the  piston  is  depressed.  A  small 
Mariotte's  tube  is  attached  to  the  receiver,  to  show  how  many 
atmospheres  have  been  admitted. 

253.  Experiments  with  the  Air  Condenser.— If  the  re- 
ceiver be  partly  filled  with  water,  and  a  pipe  from  the  stop-cock 
extend  into  it,  then  when  the  condenser  has  been  used  and  re- 
moved, and  the  stop-cock  opened,  a  jet  of  water  will  be  thrown  to 
a  height  corresponding  to  the  tension  of  the  inclosed  air.     A  gas- 
bag being  placed  in  the  condenser,  then  filled  and  shut,  will  be- 
come flaccid  when  the  air  around  it  is  compressed.     A  thin  glass 
bottle,  sealed,  will  be  crushed  by  the  same  force.     By  these  and 
other  experiments  may  be  shown  the  effects  of  increased  tension. 

254.  The  Bellows. — The  simple  or  hand-bellows  consists  of 
two  boards  or  lids  hinged  together,  and  having  a  flexible  leather 
round  the  edges,  and  a  tapering  tube  through  which  the  air  is 
driven  out.     In  the  lower  board  there  is  a  hole  with  a  valve  lying 
on  it,  which  can  open  inward.     On  separating  the  lids,  the  air  by 
its  pressure  instantly  lifts  the  valve  and  fills  the  space  between 
them  ;  but  when  they  are  pressed  together,  the  valve  shuts,  and 
the  air  is  compelled  to 

escape  through  the 
pipe.  The  stream  is 
intermittent,  passing 
out  only  when  pressure 
is  applied. 

The  compound  bel- 
lows, used  for  forges 
where  a  constant  stream 
is  needed,  are  made 
with  two  compartments. 
The  partition,  G  T 
(Fig.  171)  is  fixed,  and 
has  in  it  a  valve  V  opening  upward.  The  lower  lid  has  also  a 
valve  V  opening  upward,  and  the  upper  one  is  loaded  with  weights. 


FIG.  171. 


168 


PNEUMATICS. 


The  pipe  T  is  connected  with  the  upper  compartment.  As  the 
lower  lid  is  raised  by  the  rod  A  B,  which  is  worked  by  the  lever 
E  B,  the  air  in  the  lower  part  is  crowded  through  V  into  the 
upper  part,  whence  it  is  by  the  weights  pressed  through  the 
pipe  T  in  a  constant  stream.  When  the  lower  lid  falls,  the 
air  enters  the  lower  compartment  by  the  valve  V. 


FIG.  172. 


255.  The  Siphon.— If  .a  bent  tube  ABC  (Fig.  172)  be 
filled,  and  one  end  immersed  in  a  vessel 

of  water,  the  liquid  will  be  discharged 
through  the  tube  so  long  as  the  outer  end 
is  lower  than  the  level  in  the  vessel. 
Such  a  tube  is  called  a  siphon,  and  is 
much  used  for  removing  a  liquid  from 
the  top  of  a  reservoir  without  disturbing 
the  lower  part.  The  height  of  the  bend  B 
above  the  fluid  level  must  be  less  than  34 
feet  for  water,  and  less  than  30  inches 
for  mercury.  The  reasons  for  the  motion 
of  the  water  are,  that  the  atmosphere  is 
able  to  sustain  a  column  higher  than  E  B, 
and  that  C  B  is  longer  than  E  B.  The 
two  pressures  on  the  highest  cross-section 
B  of  the  tube  are  unequal. 

For  the  pressure  at  B  towards  the 
right  is  equal  to  the  atmospheric  pressure, 
which  call  a,  minus  the  weight  of  the 
column  E  B,  which  call  b  ;  or  P  =  a  —  b. 
The  pressure  towards  the  left  at  B  is  equal 
to  a  minus  the  weight  of  the  column  C  B, 
greater  than  E  B,  and  this  weight  we  may 

call  b  -r  c  ;  or  P'  =  a  —  (b  +  c).    The  difference  of  these  pressures 
will  determine  the  motion  at  B. 

P  -  P'  =  (a  -  b}  -  \a  -  (b  +  c} }  --=  c, 
and  this  excess  of  pressure  c  causes  a  flow  in  the  direction  E  B  C. 

The  excess  of  pressure  at  any  other  point  of  the  siphon  might 
have  been  discussed  in  the  same  general  way.  In  no  case  will 
water  flow  if  the  short  arm  exceeds  34  feet  in  length,  and  practi- 
cally it  must  be  less  than  this. 

If  the  tube  is  small,  it  may  be  filled  by  suction,  after  the  end  A 
is  immersed.  If  it  is  large,  it  may  be  inverted  and  filled,  and 
then  secured  by  stop-cocks,  till  the  end  is  beneath  the  water. 

256.  Siphon  Fountain.— In  order  that  the  flow  may  be 
maintained,  it  is  not  necessary  that  the  tube  should  contain  noth- 


SIPHON    FOUNTAIN 


169 


FIG 


FIG.  174. 


mg  but  liquid.     Air  may  collect  in  large  quantities  at  the  highest 
point  and  still  not  wholly  stop  the  action. 

Into  a  flask  F  (Fig.  173)  fit  an  air-tight  cork,  through  which 

pass  two  tubes,  one  b  entering  several  inches  into 

the  flask  and  terminating  in  a  fine  jet  a,  and  the 

other  d  c  ending  at  the  cork.     Through  the  tube  d  c 

pour  water  till  the  flask  is  filled  to  the  jet  a,  when 

inverted  as  in  the  figure.     Place  the  end  of  the  tube 

a  b  in  a  beaker  of  water  H,  and 

let  the  end  of  a  rubber  tube  lead 

from  d  to  a  pail  upon  the  floor. 

The  water  in  the  flask  will  flow 

out  through  the  tube  c  d,  and  when 

the  tension  of  the  air  in  .Fhas  been 

sufficiently  lowered,  the  pressure 

of  the  atmosphere  upon  the  water 

in  the  beaker  H  will  force  it  up 

the  tube  b  a  and  out  through  the 

jet.     The  action  will  continue,  as 

in  any  other  form  of  siphon,  so 

long  as  water  is   supplied  to  the 

short  arm. 

257.  The  Suction  Pump.— 

The  section  (Fig.  174)  exhibits  the 

construction  of  the  common  suc- 
tion pump.     By  means  of  a  lever, 

the  piston   P  is  moved  up   and 

down  in  the  tube  A   V.     In  the 

piston  is  a  valve  opening  upward, 

and  at  the  top  of  the  pipe   V  C 

is  another  valve,  shown  at  V, 
also  opening  upward.  The  latter  valve  must 
be  at  a  less  height  than  34  feet  above  the 
water  C,  the  practical  limit  being  about  29 
feet,  depending  somewhat  upon  the  weight  of 
the  valves.  When  the  piston  P  is  raised,  its  valve  is  kept  shut 
by  the  pressure  of  the  atmosphere  above.  The  air  below  the 
piston  in  the  barrel  A  V  is  rarefied  and  presses  less  and  less  upon 
the  valve  V  until  at  last  its  tension,  together  with  the  weight  of 
the  valve,  is  less  than  the  tension  of  the  air  in  the  pipe  V  C  and 
the  valve  opens,  the  air  passing  through  from  below.  Now  the  ten- 
sion of  the  air  in  V  C  being  less  than  that  of  the  atmosphere,  a 
column  of  water  will  be  forced  up  the  pipe  to  a  height  such  that 
the  tension  of  the  air  in  the  pipe  together  with  the  weight  of 
the  column  of  water  shall  equal  the  pressure  of  the  external  ;:lr. 


170  PNEUMATICS. 

When  the  upward  motion  of  P  ceases,  the  valve  V  closes  by  its 
own  weight.  When  P  descends,  on  the  return  stroke,  the  air  be- 
tween it  and  V  is  compressed  till  its  tension  is  greater  than  that 
of  the  atmosphere  and  the  weight  of  the  valve  combined,  when 
the  valve  in  P  is  raised  and  the  compressed  air  escapes.  The  pis- 
ton being  raised  again,  the  water  rises  still  higher,  till  at  length 
it  passes  through  the  valve,  and  the  piston  dips  into  it ;  after  this 
the  water  above  P  is  lifted  to  the  discharge  spout  S,  while  that  be- 
low P  is  forced  to  follow  the  piston  in  its  upward  motion  by  the 
pressure  of  the  atmosphere,  as  before. 


258.  Calculation  of  the  Force.— To  determine  the  force 
necessary  to  be  applied  to  the  piston-rod  in  pumping,  let  ua 
neglect  the  weight  of  rod,  piston,  and  valve  as  well  as  the 
friction. 

Suppose  the  water  to  have  risen  to  the  point  //  (Fig.  174).  Let 
m  =  grams  per  sq.  cm.  downward  pressure  of  the  column  If  Cr 
and  r  =  grams  per  sq.  cm.  tension  of  the  rarified  air  in  the  tube. 
These  two  tend  to  drive  the  water  downward  but  are  balanced  by 
the  atmospheric  pressure  (=  1033  grams  per  sq.  cm.)  in  the- 
cistern.  As  the  water  is  in  equilibrium  we  have  m  +  r  =  1033  ; 

.-.  r  —  1033  -  m. 

Now,  if  the  area  of  the  piston  is  Q  sq.  cm.,  its  upper  surface 
suffers  a  pressure  of  1033  Q  grams.  Upon  the  under  surface  the 
pressure  is  (Q  r  =  )  Q  (1033  —  m)  grams.  The  difference  of  these 
gives  the  lifting  force, 

F  =  [1033  Q  —  (1033  Q  —  m  Q)  =]  m  Q  grams,  that  is  to  say, 
the  lifting  force  equals  the  weight  of  a  column  of  water  whose 
cross-section  equals  the  area  of  the  piston  and  whose  height  is  H  G. 
If  Q  and  H  G  are  measured  in  cms.,  their  product  gives  the  force 
in  grams. 

Suppose  the  water  to  be  above  the  piston,  at  A.  Call  the 
downward  pressure  of  the  column  A  P,  m  grams  per  sq.  cm.,  and 
the  downward  pressure  of  the  column  P  G,  ri  grams  per  sq.  cm. 
The  pressure  upon  the  upper  side  of  P  is  Q  (1033  -t-  m'}  grams, 
and  pressure  upon  the  lower  side  is  Q  (1033  —  ri}  grams.  The 
difference  equals  the  lifting  force, 

F  =  [1033  Q  +  m'  Q  —  1033  Q  +  ri  Q]  —  Q  (m'  +  n'}  grams, 
or  is  equal  to  the  weight  of  a  column  of  water  of  cross-section  Q> 
and  height  A  C,  as  in  the  previous  case.  Fi'om  this  calculation 
we  learn  that  only  the  area  of  the  piston  and  height  of  water  in  the 
pump  above  the  surface  of  the  cistei-n  need  be  considered,  the 
diameter  of  the  pipe  V  C  not  entering  the  calculation. 


THE     FIRE-ENGINE. 


171 


259. 


The 


Forcing  Pump.—  The  piston  of  the  forcing 
pump  (Fig.  175)  is  solid,  and  the  upper  valve 
V  opens  into  the  side  pipe  VS.  In  the 
ascent  of  the  piston,  the  water  is  raised  as  in 
the  suction  pump  ;  but  in  its  descent,  a  force 
must  be  applied  to  press  the  water  which  is 
above  V  into  the  side  pipe  through  V. 

Let  P  C  =  h  cm.,  B  A  =  h'  cm.,  and  the 
diameter  of  the  piston  =  d  cm.  The  force 
expended  at  any  instant  during  the  upward 
motion  of  the  piston  is  £  IT  d?  h  grams,  and 
as  h  is  greatest  at  the  end  of  the  upward 
stroke  this  force  is  increasing.  On  the  down- 
ward stroke  the  force  is  ^  v  d*  h'  grams,  since 
the  column  P  V  balances  the  column  B  V, 
leaving  only  B  A  =  h'  to  act  ;  as  B  A  is  great- 
est at  the  end  of  the  down  stroke  this  force  is 
also  increasing. 

The  piston  is  only  one  of  many  contriv- 
ances for  producing  rarefaction  of  air  in  a 
pump-tube  ;  but  since  it  is  the  most  simple  and 
most  easily  kept  in  repair,  the  piston-pump  is 
generally  preferred  to  any  other. 

__  260.  The  Fire-Engine.—  This  machine 

~—  generally  consists  of  one  or  more  forcing 
pumps,  with  a  regulating  air-vessel,  though 
the  arrangement  of  parts  is  exceedingly  varied. 

Fig.  176  will  illustrate  the  princi- 

ples of  its  construction.      As  the 

piston,   P,   ascends,   the   water    is 

raised  through  the  valve,   V,  by  at- 

mospheric pressure.    As  P  descends, 

the  water  is  driven  through  .Pinto 

the  air-vessel,  M,    whence   by  the 

condensed  air  it  is  forced  out  with- 

out interruption  through  the  hose- 

pipe, L.     The  piston    P'  operates 

in  the  same  way  by  alternate  move- 

ments.     The    piston-rods    are  at- 

tached to  a  lever  (not  represented), 

to  which  the  strength  of  several  men 

can  be  applied  at  once  by  means  of  hand-bars  called  brakes. 

The  air-vessel  may  be  attached  to  any  kind  of  pump,  when- 

ever it  is  desired  to  render  the  stream  constant. 


FIG.  176. 


172  PNEUMATICS. 

261.  Hero's  Fountain. — The  condensation  in  the  air- 
vessel,  from  which  water  is  discharged,  may  be  produced  by  the 
weight  of  a  column  of  water.  An  illustration  is  seen  in  Hero's 
fountain,  Fig.  177.  A  vertical  column  of  water  from  the  vessel,  A, 
presses  into  the  air-vessel,  B,  and  condenses  the  air  more  or  less, 
according  to  the  height  of  A  B.  From  the  top  of  this  vessel  an 
air-tube  conveys  a  portion  of  the  compressed  air  to  a  second  air- 
vessel,  C,  which  is  nearly  full  of  water,  and  has  a  jet-pipe  rising 
from  it.  Since  the  tension  of  air  in  C  is  equal  to  that  in  J5,  a 
jet  will  be  raised  which,  if  unobstructed,  would  be  equal  in  height 
to  the  compressing  column,  A  B. 

This  plan  has  been  employed  to  raise  water  from  a  mine  in 
Hungary,  and  hence  called  "  the  Hungarian  machine." 

The  principle  of  its  application  for  this  purpose  may  be  under- 
stood from  the  annexed  diagram. 

FIG.  177. 


FIG.  178. 


Let  A  represent  a  reservoir,  or  water  supply,  situated  on  high 
ground  at  an  elevation  above  the  mouth  of  the  shaft  greater  than 
the  depth  of  shaft  to  be  drained.  From  this  reservoir  a  pipe  D 
(Fig.  178)  passes  to  the  bottom  of  a  large  and  strong  air  chamber 


MANOMETERS. 


173 


B.  From  the  top  of  the  air  chamber  a  pipe  E  passes  to  the  top 
of  a  much  smaller  chamber  C,  at  the  bottom  of  the  shaft,  from 
the  bottom  of  which  passes  the  discharge  pipe  F,  having  a  valve  at 
v.  Suppose  the  necessary  valves  to  be  supplied.  Let  all  pipes 
and  both  chambers  be  filled  with  air  only.  Open  the  valve  y, 
which  will  allow  water  from  the  mine  to  flow  into  G,  driving  out 
the  air  through  E  into  B  and  out  through  the  waste  pipe  x,  which 
must  also  be  open.  Now  close  y  and  x  and  open  D,  which  will 
permit  water  to  flow  from  A  into  B,  compressing  the  air  in  B, 
which  pressure  will  be  communicated  through  the  air-pipe  E  to 
the  surface  of  the  water  in  C,  driving  it  out  through  F.  When  B 
is  full,  or  nearly  full,  of  water,  close  D,  open  x  and  y,  and  thus 
allow  water  to  flow  into  C  and  out  of  B.  When  C  is  full  and  B 
is  empty,  repeat  the  action  as  at  first.  For  a  shaft  100  feet  deep, 
the  air  chamber  B  should  be  at  least  four  times  the  capacity  of  C. 
If  the  height  of  the  reservoir  A  above  the  mouth  of  the  shaft  be 
less  than  the  depth  of  the  shaft  to  be  drained,  the  water  must  be 
raised  by  successive  lifts. 

262.  Manometers. — These  are  instruments  for  measuring 
the  tension  of  gases  or  vapors.  The  open  manometer  or  "open 
mercurial  gauge,"  as  applied  to  the  steam  boiler,  consists  simply 
of  a  thick  glass  tube,  standing  vertical,  both  ends  open,  the  lower 
end  dipping  into  mercury  contained  in  a  closed  cistern  ;  a  pipe 
connects  the  space  above  the  mercury  in  the  cistern  with  the 
steam  space  in  the  boiler.  When  the  tension  of  the  steam  is 
equal  to  one  atmosphere  the  pressure  upon  the  mercury  in  the 
cistern  will  be  balanced  by  the  pressure  of  the  air,  transmitted 
through  the  open  upper  end  of  the  tube.  As  the  steam  pressure 
increases,  the  mercury  will  rise  in  the  tube,  at  a  pressure  of  two 
atmospheres  standing  at  about  30  inches,  at 
three  atmospheres  at  60  inches,  and  so  on.  The 
pressure  of  steam  is  always  given  as  so  many 
pounds  above  one  atmosphere  ;  a  boiler  carry- 
ing 30  Ibs.  of  steam,  really  has  45  Ibs.  internal 
pressure,  15  Ibs.  of  which  is  counterbalanced 
by  the  pressure  of  the  external  air. 

For  high  pressure  a  very  long  open  mercurial 
gauge  would  be  required  ;  in  such  cases  the 
closed  manometer,  or  closed  mercurial  gauge  may 
be  used.  This  differs  from  the  former  in  having 
the  glass  tube  A  B  closed  at  the  top,  as  repre- 
sented in  Fig.  179.  In  this  instrument  the  grad- 
uation can  be  theoretically  determined  by  Mari- 
otte's  law,  considering  that,  if  the  tube  be  of 


F  G.  179. 


174 


PNEUMATICS. 


uniform  diameter,  a  halving  of  the  length  of  the  enclosed  air  in  the 
upper  part  of  the  tube  would  correspond  to  a  doubling  of  the 
pressure  exerted  upon  it.  A  correction  must,  of  course,  be  applied 
for  the  weight  of  the  column  of  mercury  in  the  lower  part  of  the 
tube. 

As  the  uniformity  of  the  bore  of  the  tube  cannot  be  assured, 
graduation  by  comparison  with  a  standard  gauge  is  the  general 
method.  Variations,  caused  by  changes  of  temperature  of  the  en- 
closed air,  must  be  corrected  by  tables  for  the  purpose.  As  the 
graduations  crowd  together  near  the  top  of  the  tube,  as  shown  in 
the  diagram,  it  has  been  proposed  to  substitute  a  tapering,  conical 
tube  for  the  cylindrical  tube,  giving  it  such  proportions  as  to 
practically  correct  these  inequalities. 

A  metallic  gauge,  called  from  the  inventor  a  Bourdon  Gauge, 
or  some  modification  of  it,  is  in  very  common  use.  It  consists  of 
a  flattened  tube,  bent  as  in  Fig.  180,  the  closed 
end  A  being  connected  with  a  toothed  sector  €'. 
When  steam  is  admitted  through  the  stop-cock 
at  B,  the  curved  tube  tends  to  straighten  as  the 
pressure  rises,  and  the  motion  of  the  end  A  in 
the  direction  of  the  arrow  turns  the  sector  O 
about  its  axis  o,  and  by  the  teeth  gives  motion 
to  the  pinion  D,  which  carries  the  index.  These 
gauges  are  graduated  by  comparison  with  a 
standard  mercurial  gauge. 


FIG.  180. 


263.  Apparatus    for   Preserving   a   Constant    Level. — 


Let  A  B  (Fig.  181)  be  a  reser- 
voir which  supplies  a  liquid  to 
the  vessel  0  D  ;  and  suppose  it 
is  desired  to  preserve  the  level 
at  the  point  C  in  the  vessel, 
while  the  liquid  is  discharged 
from  it  irregularly  or  at  inter- 
vals. So  long  as  the  mouth  of 
the  pipe  E  is  submerged  in  the 
liquid  in  G  D,  no  air  can  enter 
the  reservoir  A  B,  and  hence  no 
liquid  can  .flow  from  it ;  but 
when  the  liquid  is  drawn  from 
G  D  so  that  the  level  G  falls, 
air  will  bubble  up  through 
the  pipe  E,  displacing  liquid  in 
A  B  till  the  end  of  the  pipe  E 


FIG.  181. 


HEIGHT    OF    THE    ATMOSPHERE.  175 

is  again  closed  ;  this  action  will  be  repeated  as  often  as  the 
level  in  G  D  falls  below  0.  The  pipe  E  should  be  of  greater 
cross-section  than  the  pipe  H,  or  else  there  must  be  a  great  head 
of  water  in  A  B,  so  that  E  may  supply  liquid  faster  than  H  can 
discharge  it. 

Problems. 

1.  The  volume  of  the  receiver  of  an  air-pump  is  E,  and  that  of 
the  barrel  is  B :  prove  that  if  the  density  of  the  air  in  the  receiver 
before  exhaustion  is  D,  the  density  after  n  complete  strokes  is, 

Dn  =  {E/(R  +  B)JnD. 

2.  The  receiver  of  an  air-pump  has  three  times  the  volume  of 
the  barrel :  find  the  density  after  ten  complete  strokes. 

3.  After  three  complete  strokes  the  density  of  the  air  in  the 
receiver  of  an  air-pump  was  to  its  original  density  as  125  :  216  : 
show  that  the  volume  of  the  receiver  was  five  times  that  of  the 
barrel. 

4.  A  pump-handle  is  1  m.  long  and  is  pivoted  at  10  cm.  from 
the  end  which  is  attached  to  the  piston-rod.     The  spout  is  5  m. 
above  the  level  in  the  cistern,  and  the  diameter  of  the  piston  is  10 
cm.     What  force  must  be  applied  to  the  end  of  the  handle  in 
order  to  maintain  a  flow  from  the  spout  ? 


CHAPTER    III. 

THE    ATMOSPHERE  — ITS    HEIGHT    AND    MOTIONS. 

264.  Virtual  Height  of  the  Atmosphere.— When  two 
fluid  columns  are  in  equilibrium  with  each  other,  their  heights 
are  inversely  as  their  specific  gravities  (Art.  194).  The  specific 
gravity  of  mercury  is  10464  times  that  of  the  air  at  the  ocean 
level.  Therefore,  if  the  air  had  the  same  density  in  all  parts,  its 
height  would  be  found  by  the  proportion, 

1  :  10464  ::2.5  :  26160  feet, 

which  is  almost  five  miles.  Hence,  the  quantity  of  the  entire  at- 
mosphere of  the  earth  is  pretty  correctly  conceived  of  when  we 
imagine  it  having  the  density  of  that  which  surrounds  us,  and 
reaching  to  the  height  of  five  miles. 


176  PNEUMATICS. 

265.  Decrease  of  Density.— But  the  atmosphere  is  very 
far  from  being  throughout  of  uniform  density.    The  great  cause 
of  inequality  is  the  decreasing  weight  of  superincumbent  air  at 
increasing  altitudes.     The  law  of  diminution  of  density,  arising 
from  this  cause,  is  the  following : 

The  densities  of  the  air  decrease  in  a  geometrical  as  the  altitudes 
increase  in  an  arithmetical  ratio.  For,  let  us  suppose  the  air  to 
be  divided  into  horizontal  strata  of  equal  thickness,  and  so  thin 
that  the  density  of  each  may  be  considered  as  uniform  throughout. 
Let  a  be  the  weight  of  the  whole  column  from  the  top  to  the 
earth,  b  the  weight  of  the  whole  column  above  the  lowest  stratum, 
c  that  of  th'e  whole  column  above  the  second,  &c.  Then  the 
weight  of  the  lowest  stratum  is  a  —  b,  and  the  weight  of  the 
second  is  b  —  c,  &c.  Now  the  densities  of  these  strata,  and  there- 
fore their  weights  (since  they  are  of  equal  thickness),  are  as  the 
compressing  forces ;  or, 

a  —  b  :  b  —  c  ::  b  :  c ; 
/.  a  c  —  b  c  =  Z>2  —  b  c  ;  /.  a  c  =  J2 ; 

.'.a  :  b  : :  b  :  c  ; 

in  the  same  way,  b  :  c  : :  c  :  d  ; 

that  is,  the  iveights  of  the  entire  columns,  from  the  successive 
strata  to  the  top  of  the  atmosphere,  form  a  geometrical  series ; 
therefore,  the  densities  of  the  successive  strata,  varying  as  the  com- 
pressing forces,  also  form  a  geometrical  series.  If,  therefore,  at  a 
certain  distance  from  the  earth,  the  air  is  twice  as  rare  as  at  the 
surface  of  the  earth,  at  twice  that  distance  it  will  be  four  times  as 
rare,  at  three  times  that  distance  eight  times  as  rare,  &c. 

By  barometric  observations  at  different  altitudes,  it  is  found 
that  at  the  height  of  three  and  a  half  miles  above  the  earth  the  air 
is  one-half  as  dense  as  it  is  at  the  surface.  Hence,  making  an 
arithmetical  series,  with  3$  for  the  common  diiference,  to  denote 
heights,  and  a  geometrical  series,  with  the  ratio  of  $,  to  denote 
densities,  we  have  the  following: 

Heights,     3£,  7,  10$,  14,  17$,  21,24$,  28,  31$,  35. 
Densities,  $,  J,  $,  £,  ^,  fa  fa,  fa,  fa,  T^£. 

According  to  this  law,  the  air,  at  the  height  of  35  miles,  is  at 
least  a  thousand  times  less  dense  than  at  the  surface  of  the  earth. 
It  has,  therefore,  a  thousand  times  less  weight  resting  upon  it ; 
in  other  words,  only  one-thousandth  part  of  the  air  exists  above 
that  height. 

266.  Actual  Height  of  the  Atmosphere.— The  foregoing 
law,  founded  on  that  of  Mariotte,  cannot,  however,  be  applicable 
except  to  moderate  distances.     If  it  were  strictly  true,  the  atmos- 
phere would  be  unlimited.    But  that  is  impossible  on  a  revolving 


MOTIONS    OP    THE    AIR.  177 

body,  since  the  centrifugal  force  must  at  some  distance  or  other 
equal  the  force  of  gravity,  and  thus  set  a  limit  to  the  atmosphere ; 
and  that  limit  in  the  case  of  the  earth  is  more  than  20,000  miles 
high.  The  actual  height  of  the  atmosphere  is  doubtless  far  below 
this  ;  for  there  can  be  none  above  the  point  where  the  repellency 
of  the  particles  is  less  than  their  weight ;  and  the  repellency  di- 
minishes just  as  fast  as  the  density,  while  the  weight  diminishes 
very  slowly.  The  highest  portions  concerned  in  reflecting  the 
sunlight  are  about  45. miles  above  the  earth.  But  there  is  reason 
to  believe  that  the  air  extends  much  above  that  height,  probably 
100  or  200  miles  from  the  earth. 

267.  The  Motions  of  the  Air. — The  air  is  never  at  rest. 
When  in  motion,  it  is  called  wind.    The  equilibrium  of  the  atmos- 
phere is  disturbed  by  the  unequal  heat  on  different  parts  of  the 
earth.     The  air  over  the  hotter  portions  becomes  lighter,  and  is 
therefore  pressed  upward  by  the  cooler  and  heavier  air  of  the  less 
heated  regions.     And  the  motions  thus  caused  are  modified  as  to 
direction  and  velocity  by  the  rotation  of  the  earth  on  its  axis. 

268.  The  Trade  Winds.— The  most  extensive  and  regular 
system  of  winds  on  the  earth  is  known  by  the  name  of  the  trade 
winds,  so  called  on  account  of  their  great  advantage  to  commerce. 
They  are  confined  to  a  belt  about  equal  in  width  to  the  torrid 
zone,  but  whose  limits  are  four  or  five  degrees  further  north  than 
the  tropics. 

In  the  northern  half  of  this  trade-wind  zone  the  wind  blows 
continually  from  the  northeast,  and  in  the  southern  half  from  the 
southeast.  As  these  currents  approach  each  other,  they  gradually 
become  more  nearly  parallel  to  the  equator,  while  between  them 
there  is  a  narrow  belt  of  calms,  irregular  winds,  and  abundant 
rains. 

The  oblique  directions  of  the  trade  winds  are  the  combined 
effects  of  the  heat  of  the  torrid  zone  and  the  rotation  of  the  earth. 
The  cold  air  of  the  northern  hemisphere  tends  to  flow  directly 
south,  and  crowd  up  the  hot  air  over  the  equator.  In  like  man- 
ner, the  cold  air  of  the  southern  hemisphere  tends  to  flow  directly 
northward.  So  that  if  the  earth  were  at  rest,  there  would  be  north 
winds  on  the  north  side  of  the  equator,  and  south  winds  on  the 
south  side.  But  the  earth  revolves  on  its  axis  from  west  to  east, 
and  the  air,  as  it  moves  from  a  higher  latitude  to  a  lower,  has  only 
so  much  eastward  motion  as  the  parallel  from  which  it  came- 
Therefore,  since  it  really  has  a  less  motion  from  the  west  than 
those  regions  over  which  it  arrives,  it  has  relatively  a  motion  from 
the  east.  This  motion  from  the  east,  compounded  witli  the  motion 
from  the  north  on  the  north  side  of  the  equator,  and  with  that 
12 


178  PNEUMATICS. 

from  the  south  on  the  south  side,  constitutes  the  northeast  and 
southeast  tradewinds. 

The  limits  of  this  system  move  a  few  degrees  to  the  north 
during  the  northern  summer,  and  to  the  south  during  the 
northern  winter,  but  very  much  less  than  might  be  expected 
from  the  changes  in  the  sun's  declination. , 

In  certain  localities  within  the  tropics  the  wind,  owing  to  pecu- 
liar configurations  of  coast  and  elevations  of  the  interior,  changes 
its  direction  periodically,  blowing  six  months  from  one  point,  and 
six  months  from  a  point  nearly  opposite.  The  monsoons  of  south- 
ern India  are  the  most  remarkable  example. 

269.  The  Return  Currents. — The   air  which  is  pressed 
upward  over  the  torrid  zone  must  necessarily  flow  away  northward 
and  southward  towards  the  higher  latitudes,  to  restore  the  equi- 
librium.    Hence,  there  are  south  winds  in  the  upper  air  on  the 
north  side  of  the  equator,  and  north  winds  on  the  south  side.    But 
these  upper  currents  are  also  oblique  to  the  meridians,  because, 
having  the  easterly  motion  of  the  equator,  they  move  faster  than 
the  parallels  over  which  they  successively  arrive,  so  that  a  motion 
from  the  west  is  combined  with  the  others,  causing  southwest 
winds  in  the  northern  hemisphere,  and  northwest  in  the  southern. 
These  motions  of  the  upper  air  are  discovered  by  observations 
made  on  high  mountains,  and  in  balloons,  and  by  noticing  the 
highest  strata  of  clouds=     It  is  to  be  borne  in  mind  that  although 
the  atmosphere  is  more  than  100  miles  high,  yet  the  lower  half 
does  not  extend  beyond  three  and  a  half  miles  above  the  earth 
(Art.  265). 

270.  Circulation  Beyond  the  Trade  Winds.— The  upper 
part  of  the  air  which  flows  away  from  the  equator  cannot  wholly 
retain  its  altitude,  because  of  the  diminishing  space  on  the  suc- 
cessive parallels.     About  latitude  30°,  it  is  so  much  accumulated 
that  it  causes  a  sensible  increase  of  pressure  (Art.  243),  and  begins 
to  descend  to  the  earth.    It  is  probable  that  some  of  the  descend- 
ing air  still  retains  its  oblique  motion  towards  higher  latitudes 
(for  the  prevailing  winds  of  the  northern  temperate  zone  are  from 
the  southwest,  and  of  the  southern  temperate  zone  from  the  north- 
west), while  a  part  joins  with  the  lower  air  which  is  moving 
towards  the  equator.     Only  so  much  of  the  rising  equatorial  mass 
can  flow  back  to  the  polar  regions  as  is  needed  to  supply  the 
comparatively  small  area  within  them.     On  account  of  the  suc- 
cessive descent  of  the  air  returning  from  the  equator,  there  is 
much  less  distinctness  and  regularity  in  the  general  circulation 
outside  of  the  torrid  zone  than  within  it.     Besides  this,  various 
local  causes,  such  as  mountain  ranges,  sea-coasts,  and  ocean  cur- 


VENTILATORS.  179 

rents,  clear  and  cloudy  skies,  &c.,  mingle  their  effects  with  the 
more  general  circulation,  and  modify  it  in  every  possible  way. 

271.  Land  and  Sea  Breezes. — These  are  limited  circula- 
tions over  adjoining  portions  of  land  and  water,  the  wind  blowing 
from  the  water  to  the  land  in  the  day  time,  and  in  the  contrary 
direction  by  night.     When  the  sun  begins  to  shine  each  day,  it 
heats  the  land  more  rapidly  than  the  water.     Hence  the  air  on 
the  land  becomes  warmer  and  lighter  than  that  on  the  water,  and 
the  surface  current  sets  toward  the  land.     By  night  the  flow  is  re- 
versed, because  the  land  cools  most  rapidly,  and  the  air  above  it 
becomes  heavier  than  that  over  the  water.     These  effects  are  more 
striking  and  more  regular  in  tropical  countries,  but  are  common 
in  nearly  all  latitudes. 

272.  A  Current  Through  a  Medium. — There  are  some 
phenomena  relating  to  currents  moving  through  a  fluid,  either  of 
the  same  or  a  different  kind,  which  belong  alike  to  hydraulics  and 
pneumatics ;  a  brief  account  of  these  is  presented  here. 

If  a  stream  is  driven  through  a  medium,  it  carries  along  the 
adjoining  particles  by  friction  or  adhesion.  The  experiment  of 
Venturi  illustrates  this  kind  of  action,  as  it  takes  place  between 
the  particles  of  water.  A  reservoir  filled  with  water  has  in  it  an 
inclined  plane  of  gentle  ascent,  whose  summit  just  reaches  the 
edge  of  the  reservoir.  A  stream  of  water  is  driven  up  this  plane 
with  force  sufficient  to  carry  it  over  the  top  ;  but  in  doing  so,  it 
takes  out  continually  some  part  of  the  water  of  the  reservoir,  and 
will  in  time  empty  it  to  the  level  of  the  lowest  part  of  the  stream. 
A  stream  of  air  through  air  produces  the  same  effect,  as  may  be 
shown  by  the  flame  of  a  lamp  near  the  stream  always  bending  to- 
ward it.  In  like  manner,  water  through  air  carries  air  with  it ; 
when  a  stream  of  water  is  poured  into  a  vessel  of  water,  air  is  car- 
ried down  in  bubbles  ;  and  cataracts  carry  down  much  air,  which 
as  it  rises  forms  a  mass  of  foam  on  the  surface.  The  strong  wind 
from  behind  a  high  waterfall  is  owing  to  the  condensation  of  air 
brought  down  by  the  back  side  of  the  sheet. 

273.  Ventilators. — If  the  stream  passes  across  the  end  of  an 
open  tube,  the  air  within  the  tube  will  be  taken  along  with  the 
stream  and  thus  a  partial  vacuum  formed,  and  a  current  estab- 
lished.    It  is  thus  that  the  wind  across  the  top  of  a  chimney  in- 
creases the  draught  within.     To  render  this  effect  more  uniformly 
successful,  by  preventing  the   wind  from  striking  the  interior 
edge  of  the   flue,  appendages,    called  ventilators,   are    attached 
to   the  chimney  top.     A  simple  one,  which  is  generally  effec- 
tual, consists  of  a  conical  frustum  surrounding  the  flue  as  in 


180  PNEUMATICS. 

Fig.  182,  so  that  the  wind,  on  striking  the  oblique  surface,  is 
thrown  over  the  top  in  a  curve,  which  is 
convex  upward.  The  same  mechanical  con- 
trivance is  much  used  for  the  ventilation  of 
public  halls  and  the  holds  of  ships.  A 
horizontal  cover  may  be  supported  by  rods, 
at  the  height  of  a  few  inches,  to  prevent 
the  rain  from  entering. 


274.  A  Stream  Meeting  a  Sur- 
face.— Though  the  moving  fluid  may  be 
elastic,  yet,  when  it  meets  a  surface,  it 
tends  to  follow  it,  rather  than  to  rebound 
from  it.  This  effect  is  partly  due  to  adhe- 
sion, and  partly  to  the  resistance  of  the 
medium  in  which  the  stream  moves.  It  will  not  only  follow  a 
plane  or  concave  surface,  but  even  one  which  is  convex,  provided 
the  velocity  of  the  current  is  not  too  great,  or  the  curvature  too 
rapid.  A  stream  of  air,  blown  from  a  pipe  upon  a  plane  surface, 
will  extinguish  the  flame  of  a  lamp  held  in  the  direction  of  the 
surface  beyond  its  edge,  while,  if  the  lamp  be  held  elsewhere  near 
the  stream,  the  flame  will  point  toward  the  stream,  according  to 
Art.  272.  Hence,  snow  is  blown  away  from  the  windward  side  of 
a  tight  fence,  and  from  around  trees. 

275.  Diminution  of  Pressure  on  a  Surface. — When  a 
stream  is  thus  moving  along  a  surface,  the  fluid  pressure  on  that 
surface  is  slightly  diminished.  This  is  proved  by  many  experi- 
ments. If  a  curved  vane  be  suspended  on  a  pivot,  and  a  stream 
of  air  be  directed  tangentially  along  the  surface,  it  will  move  to- 
ward the  stream,  and  may  be  made  to  revolve  rapidly  by  repeating 
the  blast  at  each  half  revolution.  What  is  frequently  called  the 
pneumatic  paradox  is  a  phenomenon  of  the  same  kind.  A  stream 
of  air  is  blown  through  the  centre  of  a  disk,  against  another  light 
disk,  which,  instead,  of  being  blown  off,  is  forcibly  held  near  to  it 
by  the  means.  The  pressure  is  diminished  by  all  the  radial  streams 
along  the  surface  contiguous  to  the  other  disk,  and  the  full  pressure 
on  the  outside  preponderates.  Another  form  of  the  experiment  is , 
to  blow  a  stream  of  air  through  the  bottom  of  a  hemispherical  cup, 
in  which  a  light  sphere  is  lying  loosely.  The  sphere  cannot  be 
blown  out,  but,  on  the  contrary,  is  held  in,  as  may  be  seen  by  in- 
verting the  cup,  while  the  blast  continues.  It  appears  to  be  for  a 
reason  of  the  same  sort  that  a  ball  or  a  ring  is  sustained  by  a  jet 
of  water.  It  lies  not  on  the  top,  but  on  the  side  of  the  jet,  which 
diminishes  the  pressure  on  that  side  of  the  ball,  so  that  the  air  on 
the  outside  keeps  it  in  contact.  The  tangential  force  of  the  jet 


VORTICES.  181 

causes  the  body  to  revolve  with  rapidity.    A  ball  can  be  sustained 
a  few  inches  high  by  a  stream  of  air. 

276.  Vortices  where  the  Surface  Ends.  —  As  a  current 
reaches  the  termination  of  the  surface  along  which  it  was  flowing, 
a  vortex  or  whirl  is  likely  to  occur  in  the  surrounding  medium 
behind  the  edge  of  the  surface.     Vortices  are  formed  on  water, 
whose  flow  is  obstructed  by  rocks  ;  and  often  when  the  obstruct- 
ing body  is  at  a  distance  below  the  surface,  the  whirl  which  is  es- 
tablished there  is  communicated  to  the  top,  so  that  the  vortex  is 
seen,  while  its  cause  is  out  of  sight.     There  is  a  depression  at  the 
centre,  caused  by  the  centrifugal  force  ;  and  if  the  rotation  is 
rapid,  a  spiral  tube  is  formed,  in  which  the  air  descends  to  great 
depths.     These  are  called  whirlpools.     In  a  similar  manner  whirls 
are  produced  in  the  air,  when  it  pours  off  from  a  surface.    The 
eddying  leaves  on  the  leeward  side  of  a  building  in  a  windy  day 
often  indicate  such  a  movement,  though  it  may  have  no  perma- 
nency, the  vortex  being  repeatedly  broken  up  and  reproduced. 

277.  Vortices  by  Currents  Meeting.  —  But  vortices  are 
also  formed  by  counteracting  currents  in  an  open  medium.   When 
an  aperture  is  made  in  the  middle  of  the  bottom  of  a  vessel,  as  the 
water  runs  toward  it,  the  filaments  encounter  each  other,  and 
usually,  though  not  invariably,  they  establish  a  rotary  motion, 
and  form  a  whirlpool.     Vortices  are  a  frequent  phenomenon  of 
the  atmosphere,  sometimes  only  a  few  feet  in  diameter,  in  other 
instances  some  rods  or  even  miles  in  width.     The  smaller  ones, 
occurring  over  land,  are  called   whirlwinds  ;    over  water,  water- 
spouts.   They  probably  originate  in  currents  which  do  not  exactly 
oppose  each  other,  but  act  as  a  couple  of  forces,  tending  to  produce 
rotation  (Art.  56). 

The  burning  of  a  forest  sometimes  occasions  whirlwinds,  which 
are  borne  away  by  the  wind,  and  maintain  their  rotation  for  miles. 
As  the  pressure  in  the  centre  is  diminished  by  the  centrifugal 
force,  substances  heavier  than  air,  as  leaves  and  spray,  are  likely 
to  be  driven  up  in  the  axis,  and  floating  substances,  as  cloud,  will 
for  the  same  reason  descend.  The  rising  spray  and  the  descend- 
ing cloud  frequently  mark  the  progress  of  a  vortex  in  the  air,  as 
it  moves  over  a  lake  or  the  ocean.  Such  a  phenomenon  is  called 
a  water-spout.  , 


A 


PART    IV. 


CHAPTER    I. 

NATURE  AND  PROPAGATION  OF  SOUND.   , 

278.  Sound.  —  Vibrations.  —  The  impression  which  the 
mind  receives  through  the  organ  of  hearing  is  called  sound.  But 
the  same  word  is  constantly  used  to  signify  that  progressive  vibra- 
tory movement  in  a  medium  by  which  the  impression  is  produced, 
as  when  we  speak  of  the  velocity  of  sound. 

The  vibrations  constituting  sound  are  comparatively  slow,  and 
are  often  perceived  by  sight  and  by  feeling  as  well  as  by  hearing. 
For  these  reasons,  the  true  nature  of  sound  is  investigated  with 
far  greater  ease  than  that  of  light,  electricity,  &c.  It  is  not  diffi- 
cult to  discover  that  vibrations  in  the  medium  about  us  are  essen- 
tial to  hearing  ;  and  these  vibrations  are  always  traceable  to  the 
body  in  which  the  sound  originates.  A  body  becomes  a  source  of 
sound  by  producing  an  impulse  or  a  series  of  impulses  on  the  sur- 
rounding medium,  and  thus  throwing  the  medium  itself  into 
motion.  A  single  sudden  impulse  causes  a  noise,  with  very  little 
continuance  ;  an  irregular  and  rapid  succession  of  impulses  a 
crash,  or  roar,  or  continued  noise  of  some  kind  ;  but  if  the  im- 
pulses are  rapid  and  perfectly  equidistant,  the.  effect  is  a  musical 
sound.  In  most  cases  of  the  last  kind  the  impulses  are  vibrations 
of  the  body  itself ;  and  whatever  affects  these  vibrations  is  found 
to  affect  the  sound  emanating  from  it ;  and  if  they  are  destroyed, 
the  sound  ceases. 

If  we  rub  a  moistened  finger  along  the  edge  of  a  tumbler 
nearly  full  of  water,  or  draw  a  bow  across  the  strings  of  a  viol,  we 
can  procure  sounds  which  remain  undiminished  in  intensity  as 
long  as  the  operation  by  which  they  are  excited  is  continued.  In 
both  cases  the  vibrations  are  visible  ;  those  of  the  tumbler  are 
plainly  seen  as  crispations  on  the  water  to  which  they  are  cornmu- 


AIR    AS    A    MEDIUM    OF    SOUXD.  1S3 

iricated  ;  the  string  appears  as  a  broad  shadowy  surface.  If  a  wire 
or  light  piece  of  metal  rests  against  a  bell  or  glass  receiver,  when 
ringing,  it  will  be  made  to  rattle.  If  sand  be  strewed  on  a  hori- 
zontal plate  while  a  bow  is  drawn  across  its  edge,  the  sand  will  be 
agitated,  and  dance  over  the  surface,  till  it  finds  certain  places 
where  vibrations  do  not  exist.  Near  an  organ-pipe  the  tremor 
of  the  air  is  perceptible,  and  pipes  of  the  largest  size  jar  the  seats 
and  walls  of  an  edifice.  Every  species  of  sound  may  be  traced  to 
impulses  or  vibrations  in  the  sounding  body. 

279.  Sonorous  Bodies. — Two  qualities  in  a  body  are  neces- 
sary, in  order  that  it  may  be  sonorous.  It  must  have  a  form 
favorable  for  vibrating  movements,  and  sufficient  elasticity. 

The  favorable  forms  are  in  general  rods  and  plates,  rather  than 
very  compact  masses,  like  spheres  and  cubes  ;  because  the  particles 
of  the  former  are  more  free  to  receive  lateral  movements  than 
those  of  the  latter,  which  are  constrained  on  every  side.  But  even 
.a  thin  lamina  may  have  a  form  which  allows  too  little  freedom  of 
motion,  such  as  a  spherical  shell,  in  which  the  parts  mutually  sup- 
port each  other.  If  the  shell  be  divided,  the  hemispheres  are  bell- 
shaped  and  very  sonorous. 

The  elasticity  of  some  materials  is  too  imperfect  for  continued 
vibration  ;  thus  lead,  in  whatever  form,  has  no  sonorous  quality. 

2CO.  Air  as  a  Medium  of  Sound. — There  must  not  only 
be  a  vibrating  body,  as  a  source  of  sound,  but  a  medium  for  its 
communication  to  the  organ  of  hearing.  The  ordinary  medium  is 
air.  Let  a  bell  mounted  with  a  hammer  and  mainspring,  so  as  to 
continue  ringing  for  several  minutes,  be  placed  on  a  thick  cushion 
under  the  receiver  of  an  air-pump.  The  cushion,  made  of  several 
thicknesses  of  woolen  cloth,  is  necessary  to  prevent  communica- 
tion through  the  metallic  parts  of  the  instrument.  As  the  process 
of  exhaustion  goes  on,  the  sound  of  the  bell  grows  fainter,  and  at 
length  ceases  entirely.  From  this  experiment  we  learn  that  sound 
cannot  be  propagated  through  a  vacant  space,  even  though  it  be 
only  an  inch  or  two  in  extent. 

281.  Method  of  Propagation  of  Sound. — The  vibrations  of 
a  medium  in  the  transmission  of  sound  are  of  the  kind  called  lon- 
gitudinal, i.e.,  the  particles  move  in  the  direction  of  the  propaga- 
tion. Let  Fig.  183  represent  the  arrangement  of  some  air  particles 
at  a  certain  instant.  These  particles  are  transmitting  the  sound 
vibrations  of  the  bell  at  A  to  the  point  B.  The  bell  an  instant  be- 
fore has  expanded.  In  expanding  it  has  forced  the  air  particles  in 
contact  with  it  to  come  into  collision  with  their  neighbors.  Both 


184  ACOUSTICS. 

being  perfectly  elastic,  they  exchange  velocities  and  rebound  (Art. 
96).     In  another  instant  the  motion  will  be  communicated  to  the 

FIG.  183. 
c      r    x£j£j£x  £,£..., 

i&;"%  "H  iMliiiiiliiiiii!   ill-  MM, 


next  series  of  particles.  If  the  bell  should  become  fixed  at  this  in- 
stant, the  impulse  which  it  has  just  given  would  travel  onward 
towards  B  with  a  definite  velocity,  to  be  discussed  later.  The  im- 
pulse appears  as  a  crowded  condition  of  the  particles.  On  the  other 
hand,  if  the  bell  be  not  fixed,  its  return  to  its  normal  position  will 
cause  a  rarefied  condition  of  the  air  to  follow  after  the  condensed 
condition,  with  the  same  velocity.  A  particle,  in  aiding  the  trans- 
mission of  these  two  conditions,  travels  first  towards  B,  then  back 
beyond  its  normal  position  towards  A,  and  then  returns  to  its  posi- 
tion of  equilibrium.  The  Figure  represents  an  arrangement,  re- 
sulting from  a  continued  vibration  of  the  bell,  where  c,  c',  c",  &c., 
are  condensed  conditions,  and  r,  r't  r",  &c.,  are  rarefied  condi- 
tions travelling  towards  B.  This  state'  of  affairs  constitutes  a 
wave  system,  resulting  from  the  small  vibrations  of  the  air  par- 
ticles. 

During  the  interval  of  time  that  it  takes  the  bell  to  make  a  com- 
plete vibration,  the  impulse,  which  it  first  originated,  travels  through 
a  definite  distance,  as  c  to  c',  or  r  to  r'.  Had  the  bell  vibrated 
quicker,  this  distance  would  have  been  smaller.  The  distance  be- 
tween two  successive  condensed  or  rarefied  conditions  is  termed  the 
wave  length.  As  in  water  waves  (Art.  220)  the  wave  length  is' the 
distance,  measured  on  the  line  of  propagation,  between  two  succes- 
sive particles  in  like  phases. 

The  length  of  the  path  described  by  a  single  particle  is  termed 
the  amplitude  of  vibration. 

282.  Huyghens's  Principle. — A  body  sounding  in  open  air 
sends  out  a  system  of  waves,  which  consist  of  alternate,  concentric, 
spherical  condensations  and  rarefactions.  Not  only  is  this  true, 
but,  as  Huyghens  has  shown,  each  individual  particle  sends  out 
spherical  waves  by  its  vibrating ;  it  is  owing  to  this  that  the  greater 
waves  exist.  For  instance,  a  wave  front  from  a  sounding  body,  A 


VELOCITY     OF     SOUND. 


185 


FIG.  184. 


(Fig.  184)  lias,  at  a  certain  instant,  reached  B  B.  Every  particle 
of  this  front  is  set  in  vibration  and  is  a  centre  for  further  spherical 
waves.  The  number  of  these  elementary 
spherical  waves  is  very  great,  and  they  will 
evidently  coalesce  to  form  a  new  wave  front 
B'  B'. 

This  theory  of  the  transmission  of  wave 
motions  is  called  the  Huyghens's  Principle. 
Many  phenomena  of  wave  motion  cannot  be 
adequately  explained  without  employing  it. 


283.  Velocity   of  Sound   in   Air.— 

Sound  occupies  an  appreciable  time  in  pass- 
ing through  air.  This  is  a  fact  of  common 
observation.  The  flash  of  a  distant  gun  is 
seen  before  the  report  is  heard.  Thunder 
usually  follows  lightning  after  an  interval  of 
many  seconds  ;  but  if  the  electric  discharge 
is  quite  near,  the  lightning  and  thunder  are 

almost  simultaneous.  If  a  person  is  hammering  at  a  distance,  the 
perceptions  of  the  blows  received  by  the  eye  and  the  ear  do  not 
generally  agree  with  each  other  ;  or  if  in  any  case  they  do  agree, 
it  will  be  observed  that  the  first  stroke  seen  is  inaudible,  and  the 
last  one  heard  is  invisible  ;  for  it  requires  just  the  time  between 
two  strokes  for  the  sound  of  each  to  reach  us. 

A  long  column  of  infantry,  marching  to  the  music  of  a  single 
band,  will  have  a  vertical  wave-like  motion,  since  each  rank  steps 
to  the  music,  and  a  given  beat  reaches  the  different  ranks  in  suc- 
ceeding periods  of  time. 

Many  careful  experiments  were  made  in  the  eighteenth  century 
to  determine  the  velocity  of  sound  ;  but  as  the  temperature  was 
not  recorded,  they  have  but  little  value.  During  the  present  cen- 
tur}',  the  velocity  has  been  determined  by  several  series  of  obser- 
vations in  different  countries,  and  all  reduced  to  the  same  temper- 
ature. The  agreement  between  them  is  very  close,  and  the  mean 

of  all  is  332  meters,  or  1090  feet  per  second  at  0  C. 

*. 

284.  Newton's  Formula. — Newton  showed  that  the  velocity 
of  sound  in  air  v,  could  be  calculated  from  its  elasticity  E,  and  its 
density  d  by  the  formula 

(W 


By  the  elasticity  of  the  air  is  meant  the  following :  Suppose  the 
pressure  on  the  air  were  changed  by  a  small  amount  represented 


356  ACOUSTICS. 

by  dp.  The  air  would  suffer  a  corresponding  change  of  volume 
d  v,  according  to  Marietta's  law  (Art.  237).  The  ratio  of  these  two 
is  the  elasticity,  or 

E=dP 

d  v 

A  little  consideration  will  show  that  E  increases  as  the  pressure  on 
the  air  increases.  The  density  d  also  increases  with  the  pressure, 
hence  the  velocity  of  sound  in  air  is  not  affected  by  any  change  in  the 
barometric  pressure. 

On  the  other  hand,  an  increase  of  temperature  will  increase  the 
volume  of  air,  if  it  be  under  constant  pressure,  and  hence  decreases 
its  density  d.  This  increases  the  velocity  v.  The  velocity  of  sound 
in  air  increases  about  0.6  meter  for  each  degree  Centigrade  rise  in 
temperature. 

When  the  air  particles  are  condensed,  in  transmitting  sound, 
there  is  an  increase  of  temperature,  owing  to  the  latent  heat  made 
sensible.  This  increases  the  elasticity,  and  accordingly  the  veloc- 
ity. In  the  following  rarefied  portion  of  the  wave,  cold  is  produced 
by  the  rarefaction.  This  cold  decreases  the  elasticity  of  the  rare- 
fied portion,  which  has  the  same  effect  as  increasing  the  elas- 
ticity of  the  condensed  part.  For  it  is  the  condensed  portion  that 
does  the  propagating,  the  rarefaction  *being  the  result  of  the  con- 
densation. Hence  both  the  heat  of  condensation  and  tlie  cold  of  rare- 
faction increase  the  velocity  of  sound  in  air. 

Newton's  formula  does  not  take  account  of  this  and  must  be 
accordingly  modified  by  the  introduction  of  a  constant.  We  have 
then 


285.  Velocity  as  Affected  by  the  Condition  of  the  Air 
and  the  Quality  of  the  Sound. —  Wind,  of  course,  affects  the  ve- 
locity of  sound  by  the  addition  or  subtraction  of  its  own  velocity, 
estimated  in  the  sam^e  direction,  because  it  transfers  the  medium 
itself  in  which  the  sound  is  conveyed.  This  modification,  however, 
is  only  slight,  for  sound  moves  ten  times  faster  than  wind  in  the 
most  violent  hurricane. 

But  other  changes  in  the  condition  of  the  air  produce  little  or 
no  effect.  Neither  pressure,  nor  moisture,  nor  any  change  of 
weather,  alters  the  velocity  of  sound,  though  they  may  affect  its  in- 
tensity, and  therefore  the  distance  at  which  it  can  be  heard.  Fall- 
ing snow  and  rain  obstruct  sound,  but  do  not  retard  it. 

All  kinds-  of  sound — the  firing  of  a  gun — the  blow  of  a  ham- 
mer— the  notes  of  a  musical  instrument,  or  of  the  voice,  however 


VELOCITY     IX     LIQUIDS.  187 

high  or  low,  loud  or  soft,  are  conveyed  at  the  same  rate.  That 
sounds  of  different  pitch  are  conveyed  with  the  same  velocity  was 
conclusively  proved  by  Biot,  in  Paris,  who  caused  several  airs  to 
be  played  on  a  flute  at  one  end  of  a  pipe  more  than  3000  feet  long, 
and  heard  the  same  at  the  other  end  distinctly,  and  without  the 
slightest  displacement  in  the  order  of  notes,  or  intervals  of  silence 
between  them. 

286.  Other  Gaseous  Bodies  as  Media  of  Sound.— Let 
a  spherical  receiver,  having  a  bell  suspended  in  it,  be  exhausted  of 
air  till  the  bell  ceases  to  be  heard  ;  then  fill  it  with  any  gas  or 
vapor  instead  of  air,  and  the  bell  will  be  heard  again.     By  means 
of  an  organ-pipe  blown  by  different  gases,  it  can  be  learned  with 
what  velocity  sound  would  move  in  each  kind  of  gas  experimented 
upon,  because  the  pitch  of  a  given  pipe  depends  upon  the  velocity 
of  the  waves,  as  will  be  seen  hereafter. 

From  such  experiments  the  following  velocities  at  tempera- 
ture 0°  Centigrade  have  been  deduced  : 

Carbonic  acid,  856  ft.  per  second. 

Oxygen,  1040  ft.  per  second. 

Carbonic  oxide,  1106  ft.  per  second. 

Hydrogen,  4163  ft.  per  second. 

From  Newton's  formula  the  velocity  in  gases  varies  directly  as 
the  square  root  of  the  elasticities  and  inversely  as  the  square  root 
of  the  densities  ;  hence  for  the  same  elasticities,  i.e.,  the  same 
pressures,  the  velocities  should  be  inversely  proportional  to  the 
square  root  of  the  densities.  Oxygen  being  16  times  as  heavy  as 
hydrogen,  the  velocity  of  sound  in  the  latter  should  be  four  times 
as  great  as  in  the  former,  which  conclusion  is  confirmed  by  the 
facts  given  above.  Momentary  development  of  heat  by  compres- 
sion produces,  in  all  gaseous  bodies,  the  effect  of  increasing  the 
velocity  of  sound,  as  was  shown  in  Art.  284  for  air. 

287.  Liquids  as  Media. — Many  experimenters  have  deter- 
mined the  circumstances  of  the  propagation  of  sound  in  water. 
Franklin  found  that  a  person  with  his  head  under  water  could 
hear  the  sound  of  two  stones  struck  together  at  a  distance  of  more 
than  half  a  mile.     In  1826  Colladon  made  many  careful  experi- 
ments in  the  water  of  Lake  Geneva.     The  results  of  these  and 
other  trials  are  principally  the  following  : 

1.  Sounds  produced  in  the  air  are  very  faintly  heard  by  a  per- 
son in  water,  though  quite  near.  The  kinetic  energy  of  the  light 
nir  particles  is  not  sufficient  to  give  a  large  motion- to  the  heavier 
-water  particles.  Sounds  originating  under  water  are  feebly  com- 


188  ACOUSTICS. 

municated  to  the  air  above,  and  in  positions  somewhat  oblique  are 
not  heard  at  all,  owing  to  reflection. 

2.  Sounds  are  conveyed  by  water  with  a  velocity  of  4700  feet 
per  second,  at  the  temperature  of  8.3°  C.,  which  is  more  than  four 
times  as  great  as  in  air.     The  calculated  and  the  observed  velocity 
of  sound  in  water  agree  so  nearly  with  each  other,  that  there  ap- 
pears to  be  no  appreciable  effect  arising  from  heat  developed  by 
compression. 

Calculated  velocities  are  as  follows  : 

Seine  water,  at  13°  C.  =  4714  ft.  per  second. 
30°  C.  =  5013 

Solution  of  calcic  chloride,  at  23°  C.  =  6493  ft.  per  second. 

Sulphuric  ether  at  0°  C.  =  3801  ft.  per  second. 

Hence  sound  travels  with  different  velocities  in  different 
liquids ;  the  velocity  is  greater  in  the  liquid  of  greater  density  • 
the  velocity  is  increased  by  increase  of  temperature. 

3.  Sounds  conveyed  in  water  to  a  distance,  lose  their  sonorous 
quality.     For  example,  the  ringing  of  a  bell  gives  a  succession  of 
short  sharp  strokes,  like  the  striking  together  of  two  knife-blades. 
The  musical  quality  of  the  sound  is  noticeable  only  within  600  or 
700  feet.     In  air,  it  is  well  known  that  the  contrary  takes  place  -t 
the  blow  of  a  bell-tongue  is  heard  near  by,  but  the  continued 
musical  note  is  all  that  affects  the  ear  at  a  distance. 

4.  Acoustic  shadows  are  formed  ;   that   is,   sound   passes   the 
edges  of  solid  bodies  nearly  in  straight  lines,  and  does  not  turn 
around  them  except  in  a  very  slight  degree. 

To  enable  the  experimenter  to  hear  distant  sounds  without 
placing  himself  under  water,  Colladon  pressed  down  a  cylindrical 
tin  tube,  closed  at  the  bottom,  thus  allowing  the  acoustic  pulses  in 
the  water  to  strike  perpendicularly  on  the  sides  of  the  tube.  In 
this  way,  the  faintest  sounds  were  brought  out  into  the  air.  It 
appears  to  be  true  of  sound  as  of  light,  that  it  cannot  pass  from  a 
denser  to  a  rarer  medium  at  large  angles  of  incidence,  but  suffers 
nearly  a  total  reflection. 

288.  Solids  as  Media.  —  Solid  bodies  of  high  elasticity 
are  the  most  perfect  media  of  sound  which  are  known.  An  iron 
rod — as,  for  instance,  a  lightning-rod — will  convey  a  feeble  sound 
from  one  extremity  to  the  other,  with  much  more  distinctness- 
than  the  air.  If  the  ears  are  stopped,  and  one  end  of  a  long  wire 
is  held  between  the  teeth,  a  slight  scratch  or  blow  on  the  remote 
end  will  sound  very  loud.  The  sound  in  this  case  travels  through 
the  wire  and  the  bones  of  the  head  to  the  organ  of  hearing.  The 
sound  of  earthquakes  and  volcanic  eruptions  is  transmitted  to 


MIXED    MEDIA.  18£ 

great  distances  through  the  solid  earth.  By  laying  the  ear  to  the 
ground,  the  tramp  of  cavalry  may  be  heard  at  a  much  greater  dis- 
tance than  through  the  air. 

289.  Velocity  in  Solids.  —  Structure.  —  The  velocity  of 
sound  in  cast  iron  was  estimated  by  Biot  to  be  about  11000  feet 
per  second — ten  times  greater  than  in  air.  He  obtained  this  re- 
sult by  experiments  on  the  aqueduct  pipes  in  Paris.  A  blow  upon 
one  end  was  brought  to  an  observer  at  the  other  end,  3000  feet 
distant,  both  by  the  iron  and  also  by  the  air  within  it.  The  velo- 
city in  air  being  known,  and  the  difference  of  time  observed,  the 
velocity  in  iron  is  readily  calculated. 

The  following  table  is  taken  from  Tvndall : 


NAME  op  METAL. 

At  20°  C. 

At  100°  C. 

At  200°  C. 

Lead  

4030 

3951 

Gold 

5717 

5640 

5619 

Silver 

8553 

8658 

8127 

11666 

10802 

9690 

16822 

17386 

15483 

16130 

16728 

Steel  wire 

16023 

16443 

As  a  rule  the  velocity  in  metals  decreases  with  rise  of  tempera- 
ture, but  iron  and  silver  are  shown  above  to  be  exceptions  to  this 
general  rule  between  the  limits  20°  C.  and  100°  C. 

In  one  important  particular  solids  differ  from  fluids,  namely, 
in  the  fixed  relations  of  the  particles  among  themselves.  These 
relations  are  usually  different  in  different  directions  ;  hence,  sound 
is  likely  to  be  transmitted  more  perfectly  in  some  directions  through 
a  given  solid  than  in  others.  The  scratch  of  a  pin  at  one  end  of 
a  stick  of  timber  seems  loud  to  a  person  whose  ear  is  at  the  other 
end.  The  sound  is  heard  more  perfectly  in  the  direction  of  the 
grain  than  across  it.  In  crystallized  substances  it  is  unquestion- 
ably true  that  the  vibrations  of  sound  move  with  different  speed 
and  with  different  intensity  in  the  line  of  the  axis,  and  in  a  line 
perpendicular  to  it. 

The  velocity  in  woods  along  the  fibre  is  from  about  11000  feet 
to  16000  feet ;  across  the  annual  rings  from  4500  feet  to  6000  ; 
across  the  fibre,  in  the  direction  of  the  rings  from  about  2500  feet 
to  4500  feet,  all  of  which  velocities  are  approximate  and  depend 
upon  the  wood  selected. 


290.    Mixed   Media.  —  In   all  the   foregoing  statements  it 
has  been  supposed  that  the  medium  was  homogeneous  ;  in  other 


190  ACOUSTICS. 

words,  that  the  material,  its  density,  and  its  structure,  continue 
the  same,  or  nearly  the  same,  the  whole  distance  from  the  source 
of  sound  to  the  ear.  If  abrupt  changes  occur,  even  a  few  times, 
the  sound  is  exceedingly  obstructed  in  progi'ess.  When  the  re- 
ceiver is  set  over  the  bell  on  the  pump  plate,  the  sound  in  the 
room  is  very  much  weakened,  though  the  glass  may  not  be  one- 
eighth  of  an  inch  in  thickness,  and  is  an  excellent  conductor  of 
sound.  The  vibrations  of  the  internal  air  are  very  imperfectly 
communicated  to  the  glass,  and  those  received  by  the  glass  pass 
into  the  air  again  with  a  diminished  intensity.  If  a  glass  rod  ex- 
tended the  whole  distance  from  the  bell  to  the  ear,  the  sound 
would  arrive  in  less  time,  and  with  more  loudness,  than  if  air  oc- 
cupied the  whole  extent.  For  a  like  reason,  walls,  buildings,  or 
other  intervening  bodies,  though  good  conductors  of  sound  them- 
selves, obstruct  the  progress  of  sound  in  the  air.  When  the  text- 
ure of  a  substance  is  loose,  having  many  alternations  of  material, 
it  thereby  becomes  unfit  for  transmitting  sound.  It  is  for  this  rea- 
son that  the  bell-stand,  in  the  experiment  just  referred  to,  is  set 
on  a  cushion  made  of  several  thicknesses  of  loose  flannel,  that  it 
may  prevent  the  vibrations  from  reaching  the  metallic  parts  of  the 
pump.  The  waves  of  sound,  in  attempting  to  make  their  way 
through  such  a  substance,  continually  meet  with  new  surfaces,  and 
are  reflected  in  all  possible  directions,  by  which  means  they  are 
broken  up  into  a  multitude  of  crossing  and  interfering  waves,  and 
are  mutually  destroyed.  A  tumbler,  nearly  filled  with  water,  will 
ring  clearly  ;  but  if  filled  with  an  effervescing  liquid,  it  will  lose 
all  its  sonorous  quality,  for  the  same  cause  as  before.  The  alter- 
nate surfaces  of  the  liquid  and  gas,  in  the  foam,  confuse  the  waves, 
and  deaden  the  sound. 

291.  Intensity  of  Sound.—  The  intensity  or  loudness  of  sound 
depends  upon  the  amplitude  of  the  vibrations  of  the  particles  conveying 
it.  To  obtain  a  loud  tone  from  a  piano  its  keys  must  be  struck 
with  great  force,  thus  increasing  the  amplitude  of  vibration  of  the 
strings. 

As  has  been  shown,  when  a  sound  is  produced  in  open  air  the 
wave-motion  is  propagated  in  all  directions  alike,  the  entire  system 
of  waves  around  the  point  where  sound  originates  consisting  of 
•tepherical  strata  of  air  alternately  condensed  and  rarefied.  As  the 
quantity  set  in  motion  in  these  successive  layers  increases  with  the 
square  of  the  distance,  the  amount  of  motion  communicated  to 
each  particle  must  diminish  in  the  same  ratio.  Hence,  the  inten- 
sity of  sound  varies  inversely  as  the  square  of  the  distance. 

Intensity  depends  upon  the  density  of  the  air  in  which  the 
sound  is  produced,  but  not  upon  that  of  the  air  through  which  it 


DIFFUSION     OF     SOUXD.  191 

is  transmitted.  A  sound  which  could  be  heard  in  water  at  a  dis- 
tance of  23  feet  would  be  audible  in  air  at  only  10  feet.  The  re- 
port of  a  cannon,  fired  upon  a  mountain  side,  heard  by  a  person 
in  the  rare  air  of  the  summit,  would  have  the  same  intensity  as 
the  same  report  heard  in  the  valley  below  ;  but  a  gun  fired  in  the 
rare  air  of  the  summit  might  not  be  heard  in  the  valley,  while  a 
report  in  the  valley  would  be  heard  distinctly  upon  the  summit, 
the  intensity  depending  upon  the  density  of  the  medium  in  which 
the  sound  is  produced,  as  stated  above. 

Intensity  is  modified  by  motion  of  the  air.  In  still  air  sound 
is  more  perfectly  transmitted  than  when  air  currents  exist.  In 
case  of  winds  sound  is  more  intense,  for  a  given  distance,  in  the 
direction  of  the  wind  than  in  the  contrary  direction. 

Sound  is  strengthened  by  sympathetic  vibrations  of  other  bodies 
than  that  which  first  produced  the  pulses.  A  vibrating  string  pro- 
duces a  sound  scarcely  audible  ;  but  when  it  vibrates  upon  a 
sounding  box,  the  sympathetic  vibrations  of  the  latter  are  com- 
municated to  the  air  and  a  loud  sound  results.  A  vibrating  tuning- 
fork  held  in  the  hand  cannot  be  heard  ;  the  same  fork  caused  to 
vibrate  over  the  mouth  of  a  cylinder  closed  at  one  end,  and  of  a 
length  equal  to  one-fourth  of  the  wave  length  corresponding  to  the 
pitch  of  the  fork  will  give  a  very  loud  sound.  In  all  such  cases  the 
kinetic  energy  of  the  vibrating  source  is  used  up  more  rapidly 
when  producing  the  loud  sound,  for  it  has  to  set  a  greater  mass  in 
vibration. 

292.  Diffusion  of  Sound. — Sound  produced  in  the  open  air 
tends  to  spread  equally  in  all  directions,  and  will  do  so  whenever 
the  original  impulses  are  alike  on  every  side.  But  this  is  rarely  the 
case.  In  firing  a  gun,  the  first  impulse  is  given  in  one  direction, 
and  the  sound  will  have  more  intensity,  and  be  heard  further  in 
that  direction  than  in  others.  It  is  ascertained  by  experiment, 
that  a  person  speaking  in  the  open  air  can  be  equally  well  heard 
at  the  distance  of  100  feet  directly  before  him,  75  feet  on  the  right 
and  left,  and  30  feet  behind  him  ;  and  therefore  an  audience,  in 
order  to  hear  to  the  best  advantage,  should  be  arranged  within 
limits  having  these  proportions.  But,  as  will  be  seen  hereafter, 
this  rule  is  not  applicable  to  the  interior  of  a  building. 

Sound  is  also  heard  in  certain  directions  with  more  intensity, 
and  therefore  to  a  gi'eater  distance,  if  an  obstacle  prevents  its  dif- 
fusion in  other  directions.  On  one  side  of  an  extended  wall  sound 
is  heard  further  than  if  it  spread  on  both  sides ;  still  further,  in  an 
angle  between  two  walls  ;  and  to  the  greatest  distance  of  all,  when 
confined  on  four  sides,  and  limited  to  one  direction,  as  in  a  long 
tube.  The  reason  in  these  several  cases  is  obvious ;  for  a  given 


192  ACOUSTICS. 

force  can  produce  a  given  amount  of  motion  ;  and  if  the  motion  is 
prevented  from  spreading  to  particles  in  some  directions,  it  will 
reach  more  distant  ones  in  those  directions  in  which  it  does  spread. 
Speaking-tubes  confine  the  movement  to  a  slender  column  of  air, 
and  therefore  convey  sound  to  great  distances,  and  are  on  this  ac- 
count very  useful  in  transmitting  messages  and  orders  between 
.remote  parts  of  manufacturing  edifices  and  public  houses. 


CHAPTER    II. 

REFLECTION,   REFRACTION,   AND   INFLECTION   OF   SOUND. 

293.  Reflection  of  Sound. — When  sound  waves  arrive  at  a 
boundary  between  two  media  three  things  may  occur : 

(1.)  The  particles  of  the  second  medium  may  move  with  equal 
iacility,  as  those  of  the  first.  The  wave  will  in  this  case  proceed  as 
if  the  first  medium  were  continuous. 

(2.)  The  particles  of  the  second  medium  may  move  with  less 
facility.  In  this  case  the  particles  of  the  first  medium  will  rebound. 
The  path  described  after  rebounding  will  be  the  same  as  before,  if 
the  boundary  is  perpendicular  to  it.  If  not  perpendicular,  it  will 
follow  the  laws  laid  down  in  Art.  98.  The  continuity  of  the  phases 
is  not  destroyed — simply  the  direction  of  the  line  of  propagation 
is  changed,  and  the  wave  continues  after  the  change,  and  is  said  to 
have  been  reflected. 

(3.)  The  particles  of  the  second  medium  may  move  with  greater 
facility  than  those  of  the  first.  In  this  case,  the  particles  of  a  con- 
densed portion  in  the  first  medium  may  be  said  to  have  come  into 
collision  with  less  resistance  than  they  had  expected,  and  accord- 
ingly to  have  moved  farther  than  they  intended.  Their  followers  do 
the  same,  and  the  result  is  a  reflected  wave  having  a  discontinuity 
of  phases.  The  change  in  the  direction  of  propagation  is  the  same  as 
in  case  2,  but  there  is  a  loss  of  half  a  wave  length  at  the  boundary. 

These  facts  are  as  true  for  light  waves  or  any  waves  as  for  those 
of  sound. 

Sound  waves  in  air  are  reflected  from  a  solid  barrier,  inasmuch 
as  the  air  is  perfectly  elastic,  according  to  the  law  that 

The  lines  of  propagation  of  the  incident  and  reflected  waves  make 
equal  angles  on  opposite  sides  of  the  perpendicular  to  the  reflecting 
surface. 

294.  Echoes. — When  sound  is  so  distinctly  reflected  from  a 
surface  that  it  seems  to  come  from  another  source,  it  is  called  an 


E.CHOES.  193 

echo.  Broad  and  even  surfaces,  such  as  the  walls  of  buildings  and 
ledges  of  rock,  often  produce  this  effect.  According  to  the  law 
just  given,  a  person  can  hear  the  echo  of  his  own  voice  only  by 
standing  in  a  line  which  is  perpendicular  to  the  echoing  surface. 
In  order  that  one  person  may  hear  the  echo  of  another's  voice, 
they  must  place  themselves  in  lines  making  equal  angles  with  the 
perpendicular. 

The  interval  of  time  between  a  sound  and  its  echo  enables  one 
to  judge  of  the  distance  of  the  surface,  since  the  sound  must  pass 
over  it  twice.  Thus,  if  at  the  temperature  of  23°  C.,  the  echo  of 
the  speaker's  voice  reaches  him  in  two  seconds  after  its  utterance, 
the  distance  of  the  reflecting  body  is  about  1130  feet,  and  in  that 
proportion  for  other  intervals.  Arid  he  can  hear  a  distant  echo  of 
as  many  syllables  as  he  can  pronounce  while  sound  travels  twice 
the  distance  between  himself  and  the  echoing  surface. 

The  ear  can  recognize  about  nine  successive  sounds  in  one 
second ;  two  sounds  separated  by  less  than  one-ninth  of  a  second 
Wend  and  produce  confusion ;  therefore  the  distance  from  the 
speaker  to  the  reflecting  surface  at  temperature  0°  C.  must  not 
"be  less  than  ^-°^  -^  2  =  60.5  ft.  in  order  that  an  echo  of  a  sharp 
sound  may  be  heard.  For  articulate  sound  at  ordinary  tempera- 
tures the  distance  may  be  about  112.5  ft. 

295.  Simple  and  Complex  Echoes. — When  a  sound  is 
returned  by  one  surface,  the  echo  is  called  simple ;  it  is  called 
complex  when  the  reflection  is  from  two  or  more  surfaces  at  differ- 
ent distances,  each  surface  giving  one  echo.  Thus,  a  cannon  fired 
in  a  mountainous  region  is  heard  for  a  long  time  echoed  on  all 
sides,  and  from  various  distances. 

A  complex  echo  may  also  be  produced  by  two  parallel  walls,  if 
the  hearer  and  the  source  of  sound  are  both  situated  between 
them.  The  firing  of  a  pistol  between  parallel  walls  a  few  hundred 
feet  apart  has  been  known  to  return  from  30  to  40  echoes  before 
they  became  too  faint  to  be  heard.  The  rolling  of  thunder  is  in 
part  the  effect  of  reverberation  between  the  earth  and  the  clouds. 
This  is  made  certain  by  the  observed  fact  that  the  report  of  a  can- 
non, which  in  a  level  country  and  under  a  clear  sky  is  sharp  and 
-single,  becomes  in  a  cloudy  day  a  prolonged  roar,  mingled  with 
distant  and  repeated  echoes.  But  the  peculiar  inequalities  in  the 
reverberations  of  thunder  are  doubtless  due  in  part  to  the  irregu- 
larly crinkled  path  of  the  electric  spark.  A  discharge  of  light- 
ning occupies  so  short  a  time,  that  the  sound  may  be  considered 
as  starting  from  all  points  of  its  track  at  once.  But  that  track  is 
iull  of  large  and  small  curves,  some  convex  and  some  concave  to 


;94  ACOUSTICS. 

the  ear,  and  at  a  great  variety  of  distance  ;  and  all  points  which 
are  at  equal  distances  would  be  beard  at  once.  Hence  the  origi- 
nal sound  comes  to  the  hearer  with  great  irregularity,  loud  at  one 
instant  and  faint  at  another.  These  inequalities  are  prolonged 
and  intensified  by  the  echoes  which  take  place  between  the  clouds 
and  the  earth. 

296.  Concentrated  Echoes. — The  divergence  of  sound  from 
a  plane  surface  continues  the  same  as  before,  that  is,  in  spherical 
waves,  whose  centre  is  at  the  same  distance  behind  the  plane  as 
the  real  source  is  in  front.     But  concave  surfaces  in  general  pro- 
duce a  concentrating  effect.     A  sound   originating  in   the  centre 
of  a  hollo'w  sphere  will  be  reflected  back  to  the  centre  from  every 
point  of  the  surface.     If  it  emanates  from  one  focus  of  an  ellipsoid, 
it  will,  after  reflection,  all  be  collected  at  the  other  focus.     So,  if 
two  concave  paraboloids  stand  facing  each  other,  with  their  axes 
coincident,  and  a  whisper  is  made  at  the"  focus  of  one,  it  will  be 
plainly  heard  at  the  focus  of  the   other,  though  inaudible  at  all 
points  between.     In  the  last  case  the  sound  is  twice  reflected,  and 
passes  from  one  reflector  to  the  other  in  parallel  lines.     All  these 
effects  are  readily  proved  from  the  principle  that  the  angles  of  in- 
cidence and  reflection  are  equal. 

The  speaking-trumpet  and  the  ear-trumpet  have  been  supposed 
by  many  writers  to  owe  their  concentrating  power  to  multiplied 
reflections  from  the  inner  surface.  But  a  part  of  the  effect,  and 
sometimes  the  whole,  is  doubtless  due  to  employing  the  energy  in 
one  direction,  by  preventing  lateral  diffusion,  till  the  intensity  is 
greatly  increased. 

Concave  surfaces  cause  all  the  curious  effects  of  what  are  called 
whispering  galleries,  such  as  the  dome  of  St.  Paul's,  in  London. 
In  many  of  these  instances,  however,  there  seems  to  be  a  con- 
tinued series  of  reflections  from  point  to  point  along  the  smooth 
concave  wall,  which  all  meet  simultaneously  (if  the  curves  are  of 
equal  length)  at  the  opposite  point  of  the  dome ;  for  the  whisperer 
places  his  mouth,  and  the  hearer  his  ear,  close  to  the  wall,  and  not 
in  the  fopus  of  the  curve.  The  Ear  of  Dioni/xins  was  probably  a 
curved  wall  of  this  kind  in  the  dungeons  of  Syracuse.  It  is  said 
that  the  woi'ds,  and  even  the  whispers,  of  the  prisoners  were 
gathered  and  conveyed  along  a  hidden  tube  to  the  apartment  of 
the  tyrant.  The  sail  of  a  ship  when  spread,  and  made  concave  by 
the  breeze,  has  been  known  to  concentrate  and  render  audible  to 
the  sailors  the  sound  of  a  bell  100  miles  distant. 

297.  Resonance  of  Rooms.  —  If  a  rectangular  room   has 
smooth,  hard  walls,  and  is  unfurnished,  its  reverberations  will  be 


LECTURE     ROOMS.  195 

loud  and  long-continued.  Stamp  on  the  floor,  or  make  any  other 
sudden  noise,  and  its  echoes  passing  back  and  forth  will  form  a 
prolonged  musical  note,  whose  pitch  will  be  lower  as  the  apartment 
is  larger.  This  is  called  the  resonance  of  the  room.  Now,  let 
furniture  be  placed  around  the  walls,  and  the  reverberations  will 
be  weakened  and  less  prolonged.  Especially  will  this  be  the  case 
if  the  articles  be  of  the  softer  kinds,  and  have  irregular  surfaces. 
Carpets,  curtains,  stuffed  seats,  tapestry,  and  articles  of  dress  have 
great  influence  iu  destroying  the  resonance  .of  a  room.  The  ap- 
pearance of  an  apartment  is  not  more  changed  than  is  its  resonance 
by  furnishing  it  with  carpet  and  curtains.  The  blind,  on  entering 
a  strange  room,  can,  by  the  sound  of  the  first  step,  judge  with  tol- 
erable accuracy  of  its  size  and  the  general  character  of  its  furniture. 
The  reason  why  substances  of  loose  texture  do  not  reflect  sound 
well,  is  that  they  are  not  homogeneous — the  waves  are  reflected  in 
all  dii-ections  by  successive  surfaces,  interfere  with  each  other,  and 
are  destroyed. 

293.  Halls  for  Public  Speaking. — In  large  rooms,  such  as 
churches  and  lecturing  halls,  all  echoes  which  can  accompany  the 
voice  of  the  speaker  syllable  by  syllable,  are  useful  for  increasing 
the  volume  of  sound  ;  but  all  which  reach  the  hearers  sensibly 
later,  only  produce  confusion.  It  is  found  by  experiment  that  if  a 
sound  and  its  echo  reach  the  ear  within  one-sixteenth  of  a  second 
of  each  other,  they  seem  to  be  one.  Hence,  this  fraction  of  time 
is  called  the  limit  of  perceptibility.  Within  that  time  an  echo  can 
travel  about  70  feet  more  than  the  original  sound,  and  yet  appear 
to  coincide  with  it.  If  an  echoing  wall,  therefore,  is  within  35  feet 
of  the  speaker,  each  syllable  and  its  echo  will  reach  every  hearer 
within  the  limit  of  perceptibility.  The  distance  may,  however,  be 
increased  to  40  or  even  50  feet  without  injury,  especially  if  the 
utterance  is  not  rapid.  Walls  intended  to  aid  by  their  echoes 
should  be  smooth,  but  not  too  solid  ;  plaster  on  lath  is  better  than 
plaster  on  brick  or  stone  ;  the  first  echo  is  louder,  and  the  rever- 
berations less.  Drapery  behind  the  speaker  deprives  him  of  the 
aid  of  just  so  much  echoing  surface.  A  lecturing  hall  is  improved 
by  causing  the  wall  behind  the  speaker  to  change  its  direction, 
on  the  right  and  left  of  the  platform,  at  a  very  obtuse  angle,  so 
as  to  exclude  the  rectangular  corners  from  the  room.  The  voice 
is  in  this  way  more  reinforced  by  reflection,  and  there  is  less 
resonance  arising  from  the  parallelism  of  opposite  walls.  Panel- 
ing, and  any  other  recesses  for  ornamental  purposes,  may  exist  in 
the  reflecting  walls  without  injury,  provided  they  are  not  curved. 
The  ceiling  should  not  be  so  high  that  the  reflection  from  it 
would  be  delayed  beyond  the  limit  of  perceptibilit}'.  Concave 


106 


ACOUSTICS. 


surfaces,  such  as  domes,  vaults,  and  broad  niches,  should  be  care- 
fully avoided,  as  their  effect  generally  is  to  concentrate  all  the 
sounds  they  reflect.  An  equal  diffusion  of  sound  throughout  the 
apartment,  not  concentration  of  it  to  particular  points,  is  the  ob- 
ject to  be  sought  in  the  arrangement  of  its  parts. 

As  to  distant  parts  of  a  hall  for  public  speaking,  the  more  com- 
pletely all  echoes  from  them  can  be  destroyed,  the  more  favorable 
is  it  for  distinct  hearing.  It  is  indeed  true  that  if  a  hearer  is 
within  35  feet  of  a  wall,  however  remote  from  the  speaker,  he  will 
hear  a  syllable,  and  its  echo  from  that  wall,  as  one  sound ;  but  to 
all  the  audience  at  greater  distances  from  the  same  wall,  the  echoes 
will  be  perceptibly  retarded,  and  fall  upon  subsequent  syllables, 
thus  destroying  distinctness.  The  distant  walls  should,  by  some 
means,  be  broken  up  into  small  portions,  presenting  surfaces  in 
different  directions.  A  gallery  may  aid  in  effecting  this  ;  and  the 
seats  of  the  gallery  and  of  the  lower  floor  may  rise  rapidly  one 
behind  another,  so  that  the  audience  will  receive  directly  much  of 
the  sound  which  would  otherwise  go  to  the  remote  wall,  and  be 
reflected.  Especially  should  no  large  and  distant  surfaces  be 
parallel  to  nearer  ones,  since  it  is  between  parallel  walls  that  pro- 
longed reverberation  occurs. 

299.  Refraction  of  Sound.  —  It  has  been  ascertained  by 
experiment  that  sound,  like  light,  may  be  refracted,  or  bent  out  of 
its  rectilinear  course  by  entering  a  substance  of  different  density. 
If  a  large  convex  lens  be  formed  of  carbonic  acid  gas,  by  inclosing 
it  in  a  sphere  of  thin  india-rubber,  a  feeble  sound,  like  the  ticking 
of  a  watch,  produced  on  one  side,  will  be  concentrated  to  a  focal 
point  on  the  other.  In  this  case,  the  several  diverging  rays  of 
sound  are  refracted  toward  each  other  on  entering  the  sphere,  and 
still  more  on  leaving  it,  so  that  they  are  converged  to  a  focus. 


If  air- waves  are  allowed  to  pass 
FIG.  185. 


300.  Inflection  of  Sound. 
through  an  opening  in  an 
obstructing  Avail,  they  are 
not  entirely  confined  with- 
in the  radii  of  the  wave- 
system  produced  through 
the  opening,  but  spread 
with  diminished  intensity 
in  lateral  directions.  The 
particles  near  the  edges  of 
the  opening,  as  B  and  C 
(Fig.  185)  are  (Art.  282) 
sources  of  sound  ;  and  if  they  be  made  centres  of  concentric 


CHARACTERISTICS     OF     MUSICAL    SOUNDS.       197 

spheres,  whose  radii  are  equal  to  the  length  of  the  wave,  B  b,  or  Cc, 
and  its  multiples,  then  these  spherical  surfaces  will  represent  the 
lateral  systems  of  waves  which  are  diffused  on  every  side  of  the 
direct  beam,  B  D,  C  E.  But  the  sound  is  in  general  more  feeble  as 
the  distance  from  B  D,  or  C  E,  is  greater,  and  in  certain  points  is 
destroyed  by  interference.  This  spreading  of  sound  in  lateral  di- 
rections is  called  the  inflection  of  sound. 


CHAPTER    III. 

MUSICAL    SOUNDS    AND    MODES   OF    PRODUCING  THEM. 

301.  Characteristics  of  Musical  Sounds. — When  the  im- 
pulses of  a  sounding  body  upon  the  air  are  equiperiodic,  and  of  suf- 
ficient frequency,  they  produce  what  is  termed  a  musical  sound. 
In  most  cases  these  impulses  are  the  isochronous  vibrations  of  the 
body  itself,  but  not  necessarily  so  ;  it  is  found  by  experiment  that 
blows  or  pulses,  of  any  species  whatever,  if  they  are  more  than 
about  15  or  20  per  second ,  and  possess  the  property  of  isochronism, 
•cause  a  musical  tone.     For  example,  the  snapping  of  a  stick  on 
the  teeth  of  a  metallic  wheel  would  seem  as  unlikely  as  anything 
to  produce  a  musical  sound  ;  but  when  the  wheel  is  in  rapid  mo- 
tion, the  succession  causes  a  pure  musical  note. 

Musical  tones  possess  three  fundamental  independent  char- 
acteristics depending  upon  the  character  of  the  vibrations  of  the 
sounding  body : 

(1.)  Pitch,  depending  on  the  frequency  of  vibrations. 

(2.)  Intensity  or  Loudness,  depending  on  the  amplitude  of  the 
vibrations. 

(3.)  Quality  or  Timbre,  depending  on  the  form  or  shape  of  the 
vibrations. 

302.  The  Pitch  of  Musical  Sounds. — What  is  called  the 
pitch  of  a  musical  sound,  or  its  degree  of  acuteness,  is  owing  en- 
tirely to  its  rate  of  vibration.     In  comparing  one  musical  sound 
with  another,  if  the  number  of  vibrations  per  second  is  greater,  the 
sound  is  more  acute,  and  is  said  to  be  of  a  higher  pitch  ;  if  the  vi- 
brations are  fewer  per  second,  the  sound  is  graver,  or  of  a  lower 
pitch. 

303.  The  Monochord  or  Sonometer. — If  a  string  of  uni- 
form size  and  texture  is  stretched  on  a  box  of  thin  wood,  by  means 
of  a  pulley  and  weight,  the  instrument  is  called  a  monochord,  and 


198  ACOUSTICS. 

is  useful  for  studying  the  laws  of  vibrations  in  musical  sounds. 
The  sound  emitted  by  the  vibrations  of  the  whole  length  of  the 
string  is  called  its  fundamental  sound. 

If  the  string  be  drawn  aside  from  its  straight  position,  and 
then  released,  one  component  of  the  force  of  tension  urges  every 
particle  back  towards  its  place  of  rest ;  but  the  string  passes  be- 
yond that  place,  on  account  of  the  momentum  acquired,  and  de- 
viates as  far  on  the  other  side  ;  from  which  position  it  returns,  for 
the  same  reason  as  before,  and  continues  thus  to  vibrate  till  ob- 
structions destroy  its  motion.  By  the  use  of  a  bow,  the  vibrations 
may  be  continued  as  long  as  the  experimenter  chooses- 

The  pitch  of  the  fundamental  sound  of  musical  strings  is  found 
by  experience  to  depend  on  three  circumstances ;  the  length  of  the 
string — its  weight  or  quantity  of  matter — and  its  tension.  The 
tone  becomes  more  acute  as  we  increase  the  tension,  or  diminish 
either  the  length  or  the  weight.  The  operation  of  these  several 
circumstances  may  be  seen  in  a  common  violin.  The  pitch  of  any 
one  of  the  strings  is  raised  or  lowered  by  turning  the  screw  so  as 
to  increase  or  lessen  its  tension ;  or,  the  tension  remaining  the 
same,  higher  or  lower  notes  are  produced  by  the  same  string,  by 
applying  the  fingers  in  such  a  manner  as  to  shorten  or  lengthen 
the  string  which  is  vibrating  ;  or,  both  the  tension  and  the  length 
of  the  string  remaining  the  same,  the  pitch  is  altered  by  making 
the  string  larger  or  smaller,  and  thus  increasing  or  diminishing- 
its  weight. 

304.  Time  of  a  Complete  Vibration. — The  mathematical 
formula  for  the  time  of  a  complete  vibration  is 


in  which  T  is  the  time,  in  seconds,  of  a  vibration  ;  I  —  length  of 
the  string  ;  w  =  the  weight  of  a  unit  length  of  the  string  ;  t  =  the 
tension,  and  g  =  the  acceleration  of  gravity.  In  applying  the 
formula  I  and  g  must  be  in  the  same  unit,  either  both  in  centi- 
meters or  both  in  feet,  and  also  w  and  t  in  the  same  unit,  both  in 
grams  or  both  in  pounds. 

The  constant  factors,  g  and  2  being  omitted,  the  variation  may 
be  expressed  thus  : 

T*1-..  that  is, 


The  time  of  a  vibration  varies  as  the  length  of  the  string  multi- 
plied by  the  square  root  of  its  weight  per  unit  length,  and  divided  by 
the  square  root  of  its  tension. 

As  the  distance  of  the  string  from  its  quiescent  position  does 


HARMONICS.  199 

not  form  an  element  of  the  algebraic  expression  for  the  time  of  a 
vibration,  it  follows  that  the  time  is  independent  of  the  amplitude. 
Hence,  as  in  the  pendulum,  the  vibrations  of  a  string,  fixed  at 
both  ends,  are  performed  in  equal  times,  whether  the  amplitude 
of  the  vibrations  be  greater  or  smaller.  It  is  on  this  account  that 
the  pitch  of  a  string  does  not  alter,  when  left  to  vibrate  till  it 
stops.  The  excursions  from  side  to  side  grow  less,  and  therefore 
the  sound  more  feeble,  till  it  ceases  ;  but  the  rate  of  vibration,  and 
therefore  the  pitch,  remains  the  same  to  the  last.  This  property 
of  isochronism,  independent  of  extent  of  excursion,  is  common  to 
sounding  bodies  generally. 

305.  The  Number  of  Vibrations  in  a  Given  Time. — The 

greater  is  the  time  of  one  vibration,  the  less  will  be  the  number 
of  vibrations  in  a  given  time  ;  that  is,  if  N  represents  the  number, 

1  ^  t  1 

N  oc  -— ,  .-.  N  oc  - — — -•     If  t  and  w  are  constant,  N  oc  -;  if  I  and  t 

J.  i  V  w  t 

are  constant,  N  oc  ;  and  if  I  and  w  are  constant,  N  oc  \/y. 

v   iv 
that  is, 

1.  The  number  of  vibrations  varies  inversely  as  the  length. 

2.  Tlie  number  of  vibrations  varies  inversely  as  the  square  root  of 
.the  weiglit  of  the  string. 

3.  The  number  of  vibrations  varies  as  the  square  root  of  the 
tension. 

Thus,  the  number  of  vibrations  in  a  second  may  be  doubled, 
either  by  halving  the  length  of  the  string,  or  by  making  its  weight 
one-fourth  as  great,  or,  finally,  by  making  its  tension  four  times  as 
great. 

306.  Vibrations  of  a  String  in   Parts.— The  string  of  a 
monochord  may  be  made  to  vibrate  in  parts,  the  points  of  division 
remaining  at  rest ;  and  this  mode  of  vibration  may  even  coexist 
with  the  one  already  described.     Of  course  the  sound  produced 
by  the  parts  will  be  on  a  higher  pitch,  since  they  are  shorter,  while 
the  tension  and  the  weight  per  unit  length  remain  unaltered.     It 
is  a  noticeable  fact  that  the  parts  are  always  such  as  will  exactly 
measure  the  whole  without  a  remainder.     Hence  the  vibrating 
parts  are  either  halves,  thirds,  fourths,  or  other  aliquot  portions. 
The  sounds  produced  by  any  of  these  modes  of  vibration  are  called 
harmonics,  for  a  reason  which  will  appear  hereafter.     Suppose  a 
string  (Fig.  186)  to  be  stretched  between  A  and  B,  and  that  it  is 
thrown  into  vibration  in  three  parts.     Then  while  A  D  makes  its 
excursion  on  one  side,  D  C  will  move  in  the  opposite  direction, 
and  C  B  the  same  as  A  D ;  and  when  one  is  reversed,  the  others 


200  ACOUSTICS. 

are  also,  as  shown  by  the  dotted  line.  In  this  way  D  and  C  are' 
kept  at  rest,  being  urged  toward  one  side  by  one  portion  of  string, 
and  toward  the  opposite  by  the  next  portion.  But  the  string- 
may  at  the  same  time  vibrate  as  a  whole  ;  in  which  case  D  and  C 
will  have  motion  to  each  side  of  their  former  places  of  rest,  while 
relatively  to  them  the  three  portions  will  continue  their  movements 

FIG.  186. 


as  before.  The  points  C  and  D  are  called  nodes ;  the  parts  A  Dr 
D  C,  and  C  B,  are  called  ventral  segments.  By  a  little  change  in 
the  quickness  of  the  stroke,  the  bow  may  be  made  to  bring  from 
the  mouochord  a  great  number  of  harmonic  notes,  each  being  due 
to  the  vibrations  of  certain  aliquot  parts  of  the  string.  By  confin- 
ing a  particular  point,  however,  at  the  distance  of  £,  -J,  or  other 
simple  fraction  of  the  whole  from  the  end,  the  particular  harmonic 
belonging  to  that  mode  of  division  may  be  sounded  clear,  and  un- 
mingled  with  the  others. 

307.  Longitudinal  Vibrations  of  Strings. — A  string  may 
vibrate  in  the  direction  of  its  length,  in  consequence  of  its  trac. 
tional  elasticity  (Art.  100).  Such  vibrations  can  be  produced  by 
rubbing  the  stretched  string  quickly  with  resined  leather  in  the 
direction  of  its  length. 

The  fundamental  note  thus  produced  is  of  much  higher  pitch: 
than  that  of  the  same  string  caused  to  vibrate  transversely. 

By  experimenting  as  in  the  case  of  transverse  vibrations  it  may 
be  shown  that  the  number  of  vibrations  in  a  given  time  is  inversely 
proportional  to  the  length  of  the  string. 

By  altering  the  tension  within  the  limits  of  elasticity  of  the 
substance  no  change  of  pitch  is  produced,  longitudinal  vibrations 
differing  in  this  respect  from  transverse. 

Changing  the  thickness  or  weight,  the  material  being  the  same, 
does  not  alter  the  pitch,  another  difference  to  be  noted  between- 
transverse  and  longitudinal  vibrations. 

Two  wires  of  different  material  but  of  the  same  length  will  give 
different  notes. 

Now  if  two  wires  of  the  same  length  but  of  different  materials 
be  used,  that  which  transmits  *the  sound  pulse  with  the  greatest 
velocity  will  give  the  highest  note,  since  the  number  of  vibrations 
per  second  due  to  this  greater  velocity  of  transmission  will  be 
greater.  If  two  wires  of  different  materials  be  so  adjusted  as  to> 


RESONANT     CAVITIES.  201 

length  as  to  give  the  same  note,  the  ratio  of  their  lengths  is  the 
ratio  of  the  velocities  of  transmission  of  sound  in  the  two  sub- 
stances. 

308.  Pitch  or  Frequency  determined  by  Wave  Length. 
— We  have  seen  that  air,  while  transmitting  sound,  is  arranged  in 
alternate   layers   of 
condensation     and 
rarefaction.     These 


187,  the  sounding 
body  being  a  tun- 
ing-fork. The  dis- 
tance between  two 

successive  condensations,  as  c  c',  is  called  a  wave  length.  This 
wave  length  depends  upon  two  things-^the  velocity  of  sound  propa- 
gation in  air  and  the  frequency  of  the  vibrations  at  the  source. 
Now,  as  the  velocity  of  propagation  is  nearly  constant,  the  wave 
length  may  be  taken  as  a  measure  of  the  frequency  or  pitch.  The 
shorter  the  wave  length  the  higher  the  pitch.  The  same  pitch 
will  always  have  the  same  wave  length,  whatever  be  the  sound- 
ing source.  If,  for  instance,  the  prongs  of  a  tuning-fork,  giving 
256  vibrations  per  second,  move  away  from  each  other,  they  will 
produce  a  condensation  of  air  in  their  vicinity.  This  condensa- 
tion will  travel  away  from  the  fork  to  a  distance  of  1090  feet  in  1 
second,  providing  the  temperature  be  0°  C.  In  the  meantime,  how- 
ever, the  fork  has  vibrated  and  has  sent  out  255  other  condensa- 
tions. These  follow  each  other,  being  separated  by  equal  distances. 
Evidently  1090  divided  by  this  distance  equals  256,  or  the  frequency 
of  the  vibrations  of  the  sounding  body.  The  wave  length  is  then 
about  4.26  feet,  or  about  52  inches. 

309.  Resonant  Cavities. — If  the  fork,  just  employed,  be  ex- 
cited and  held  over  the  top  of  a  15-inch  jar  (Fig.  188),  and  if  water 
be  slowly  poured  into  the  jar,  it  will  be  found  that  when  the  water 
has  reached  a  certain  height  the  loudness  of  the  sound  given  off 
by  the  fork  will  be  greatly  increased.  This  reinforcement  is  caused 
by  the  vibration  of  the  air  column  above  the  water.  It  seems  that 
air  is  not  only  adapted  to  transmit  sound,  but  also  to  serve  as  a 
source  of  sound.  If,  now,  more  water  be  poured  in,  the  reinforce- 
ment will  cease.  It  appears  that  a  column  of  air  of  definite  length 
is  capable  of  responding  to,  or  being  set  in  vibration  by,  a  sound- 
ing body  having  a  52-inch  wave  length.  Measurement  will  show 
that  this  is  13  inches.  Had  a  fork  of  different  pitch  been  used,  the 
length  of  the  resonant  air  column  would  have  been  different.  In 


202 


ACOUSTICS. 


FIG.  188. 


each  case,  however,  the  length  of  the  air  column  would  be  one- 
quarter  of  the  wave  length  to 
which  it  responds. 

In  order  to  understand 
the  relation  existing  between 
wave  lengths  and  the  lengths 
of  responding  resonators  let 
us  refer  to  Fig.  189.  The 
lower  prong  of  the  fork  re- 
quires -%fa  of  a  second  to 
move  from  the  position  a  to 
b.  During  this  interval  the 
pulse  of  condensation,  caused 
by  the  initial  movement  at  a, 
•would  travel  through  26  inch- 
es, if  the  fork  were  in  open 
air.  In  the  resonating  jar  the 
pulse  will  ti'avel  through  the 
same  distance,  but  will  be  re- 
flected at  the  surface  of  the 
water.  It  will  move  through  13  inches,  be  reflected  at  the 
water,  and  retrace  the  13  inches,  arriving  at  b  in  time  to  coincide 
in  direction  with  the  movement  of  the  prong  back  to  o.  In  the 
same  manner,  the  rarefied  impulse,  caused  by 
the  upward  movement  of  the  prong,  will  trav- 
erse the  air  column  and  return  to  a  at  the  proper 
instant.  The  whole  air  column  vibrates  in  the 
same  time  as  the  fork,  and,  acting  as  an  addi- 
tional source  of  sound,  increases  the  loudness. 

310.  Stationary  Sound  Waves.— The  vi- 
brations of  the  air  particles  iu  a  resonator  are 
said  to  form  stationary  wave*,  i.e.,  some  of  the 
particles  which  convey  the  wave  do  not  move.  The  position  of 
these  particles  is  termed  a  node.  In  the  case  of  the  resonator  just 
discussed,  the  reflecting  water  surface  is  a  node,  for  the  air  par- 
ticles in  its  proximity  do  not  move  when  transmitting  the  wave. 
For  instance,  suppose  a  condensation  is  suffering  reflection.  The 
incoming  portion  of  the  wave  tends  to  crowd  the  particles  into  the 
water,  but  the  reflected  portion  tends  to  drive  the  particles  away 
from  the  water,  and,  under  their  combined  efforts,  the  particles  re- 
main at  rest.  They  are  equally  stationary,  whatever  be  the  phase 
of  the  wave  undergoing  reflection.  The  particles  at  the  open  por- 
tion of  the  resonator,  however,  suffer  a  maximum  displacement  and 
correspond  to  the  ventral  segments  of  vibrating  strings  (Art.  30(5). 


FIG.  189. 


ORGAN     PIPES.  203 

311.  Stopped  Organ  Pipes. — If  a  stream  of  air  be  blown 
over  the  top  of  our  resonator,  a  tone  of  256  vibrations  per  second 
•will  result.     A  fluttering  of  the  air  particles  at  the  open  end  is 
caused  by  the  stream,  and  the  whole  air  column  picks  out 

those  movements  which  are  synchronous  with  its  own  rate, 
and  allows  them  to  set  it  into  vibration.  Stopped  mouth 
organ  pipes  ai*e  constructed  on  this  principle.  Fig.  190 
represents  a  section  of  such  a  pipe.  Wind  from  a  wind- 
chest  entering  the  channel  i  and  issuing  at  the  mouth  o 
strikes  against  the  sharp  lip  b.  The  sharp  edge  causes  a 
chattering  much  the  same  as  an  oar  causes  ripples  in  a  river 
current.  The  air  column  in  the  pipe  is  thus  set  in  mo- 
tion and  gives  a  tone  whose  wave  length  is  four  times  the 
length  of  the  column.  That  it  is  the  air,  and  not  the  pipe 
itself,  which  is  the  source  of  sound,  is  proved  by  using 
pipes  of  various  materials — the  most  elastic  and  the  most 
inelastic — as  glass,  wood,  paper,  and  lead  ;  if  they  are  of 
the  same  form'  and  size,  the  tone  has,  in  each  case,  the 
same  pitch. 

312.  Open  Organ  Pipes. — If  the  stopped  end  of  a 
stopped  pipe  be  removed  (Fig.  191)  it  will  still  give  a  musical  tone. 
The  wave  length,  however,  will  be  but  half  what  it  was  before. 
The  pitch  shows  this.     The  sound  is  caused  by  the  sta- 
tionary waves  of  the  air  column,  and,  as  both  ends  of    FlG-  191- 
the  column  are  in  contact  with  the  open  air,  these  parts 

are  ventral  segments.  Bet\veen  these  points  there  must 
be  at  least  one  node.  In  case  the  pipe  is  giving  its  fun- 
damental tone  there  is  one  node  only,  and  this  is  half- 
way up  the  pipe. 

313.  Vibrations  of  a  Column  of  Air  in  Parts— 
Nodes. — If  air  enters  an  organ  pipe  under  a  much 
greater  pressure  than  that  intended  for  the  pipe,  this 
will  give  a  tone  of  higher  pitch  than  its  fundamental 
tone.    If  the  pressure  be  now  gradually  reduced,  a  point 
will  be  reached  when  the  pipe  will  yield  the  two  tones 
simultaneously.     The  explanation  of  this  is  that  a  col- 
umn of  air  can  vibrate  in  parts  the  same  as  a  string  (Art.  306). 
Also  it  can  vibrate  as  a  whole  and  in  parts  at  the  same  time. 

In  the  case  of  a  stopped  pipe  the  vibrations  are  limited  only  in 
that  the  stopped  end  must  always  be  a  node  and  the  other  end 
must  be  a  ventral  segment ;  a  node  because  no  motion  of  the  air 
particles  is  possible  at  the  stop,  and  a  ventral  segment  because 


204 


ACOUSTICS. 


FIG.  192.  FIG.  193. 
K 


f 


f 


FIG.  196. 


all  vibrations  are  generated  at  the  mouth.  When  a  stopped  pipe 
gives  its  fundamental  tone  it  has  but  one  node  and  one  ventral 
segment.  The  next  higher  tone  which  can  be 
produced  has  three  times  the  frequency  of  vibra- 
tion, for  the  necessity  that  the  top  Vemain  a- 
node  and  the  bottom  a  ventral  segment  requires 
the  insertion  of  one  extra  node  and  one  extra  ven- 
tral segment.  This  division  is  shown  in  Fig.  192, 
where  nodes  and  segments  are  represented  by  N~ 
and  V  respectively.  The  wave  length  has  been 
reduced  to  one-third  its  former  value.  The  next 
forced  tone  requires  the  insertion  of  two  nodes 
and  two  segments.  Their  arrangement  is  shown  in 
Fig.  193.  The  wave  length  is  one-fifth  that  of  the 
fundamental.  Thus  it  can  be  seen  that  a  stopped 
pipe  can  be  forced  to  give  tones  whose  frequencies 
of  vibrations  are  as  the  odd  numbers,  1,  3,  5,  &c. 

In  the  case  of  an  open  pipe,  both  ends  must 
remain  ventral  segments.  The  first  forced  tone  requires 
that  the  arrangement  shall  be  as  shown  in  Fig.  194. 
FIG.  194.  FIG.  195.  The  wave  length  is  half  its  funda- 
mental value,  and  the  frequency  is 
accordingly  doubled.  The  next  forc- 
ing gives  one-third  the  original  wave 
length  (Fig.  195)  and  the  frequency 
is  trebled.  An  open  pipe  can  be 
forced  to  yield  tones  whose  frequencies 
are  as  1,  2,  3,  4,  &c.  The  position  of 
nodes  in  pipes  may  be  readily  shown 
by  introducing  a  ring,  R  (Fig.  194), 
over  which  is  stretched  a  thin  mem- 
brane. If  this  ring,  suspended  from  R 
a  cord,  be  lowered  into  the  open  up- 
~7  r/"  '  per  end  of  a  sounding  pipe  of  glass, 

u)/  ill/  or  one  wu*cn   uas  one  transparent 

•"  face,  it  will  make  a  rattling  or  flut- 

tering noise,  or  will  show  that  it  is  vibrating  by  the 
motion  of  grains  of  sand  sprinkled  upon  it ;  this  vibra- 
tion decreases  in  intensity  as  the  disc  is  gradually 
lowered,  until,  when  the  disc  reaches  the  place  of  a 
node,  the  vibration  ceases  and  the  sand  remains  at 
rest.  Thus  the  place  of  each  of  any  number  of  nodes 
may  be  determined  experimentally.  Fine  silica  pow- 
der in  a  sounding  pipe  held  horizontally  will  also  mark  the  seg- 
ments and  nodes  very  beautifully. 


A 


MOUTH     AND     REED     PIPES. 


205 


FIG.  19&. 


FIG.  191; 


314.  Modes  of  Exciting  Vibrations  in  Pipes. — There  are 
two  methods  of  making  the  air  column  in  a  pipe  to  vibrate  :  one 
by  a  stream  of  air  blown  across  an  orifice  in  the  pipe,  the  other  by 
an  elastic  plate  called  a  reed.  A  familiar  example  of  the  first  is 
the  flute.  A  stream  of  air  from  the  lips  is  directed  across  the  em- 
bouchure, so  as  just  to  strike  the  oppo- 
site edge  ;  this  excites  a  wave  in  the 
tube.  A  large  proportion  of  the  pipes 
of  an  organ  are  made  to  produce  musi- 
cal tones  essentially  in  the  same  way  as 
the  flute,  and  are  called  mouth-pipes. 

Reeds  are  of  two  kinds,  the  free 
reed  and  the  beating  reed.  The  former 
passes  to  and  fro  through  an  aperture 
without  touching  its  sides.  Harmoni- 
ums and  accordeons  employ  free  reeds. 
The  beating  reed  falls  against  the  sides 
of  an  aperture,  which  it  periodically 
opens  and  closes.  The  clarinet  is  ex- 
cited by  such  a  one.  In  that  instru- 
ment the  reed  is  often  made  of  wood  ; 
when  the  air  is  blown  past  its  edge  into 
the  tube,  the  reed  is  thrown  into  vibra- 
tion, and  by  it  the  column  of  air.  What 
are  called  the  reed  pipes  of  the  organ 
are  constructed  on  the  same  principle, 
but  the  reeds  are  metallic.  An  ex- 
ample is  seen  in  Fig.  197,  which  repre- 
sents a  model  of  the  reed  pipe,  made 
to  show  the  vibrations  through  the 
glass  walls  at  E.  A  chimney,  H,  is  usually  attached,  sometimes  of 
a  form  (as  in  the  figure)  to  increase  the  loudness  of  the  sound,  and 
sometimes  of  a  different  form,  for  softening  it. 

The  lips  of  the  player  act  as  a  reed  to  the  cornet  or  trombone. 
These  instruments  are  open  pipes,  and,  by  depressing  the  valves, 
the  length  of  the  pipe  is  altered,  the  wave  being  made  to  traverse 
side  channels  in  addition  to  its  ordinary  path. 

The  air  in  an  open  tube  may  also  be  thrown  into  vibration  by 
a  burning  jet  of  gas,  as  in  Fig.  198.  The  pitch  depends  upon  the 
size  of  the  flame  and  the  length  of  the  tube.  By  varying  the  posi- 
tion of  the  jet  in  a  long  tube  a  series  of  frequencies  in  the  ratio  of 
1:2:3:4,  &c.,  can  be  obtained. 

If  a  gauze  diaphragm  be  inserted  in  a  tube  open  at  both  ends, 
about  two  inches  in  diameter  and  two  feet  long,  at  a  point  three  or 


206  ACOUSTICS. 

four  inches  from  the  end,  and  this  diaphragm  be  heated  red  hot 
by  a  Bunsen  burner,  upon  removing  the  burner  and  depressing 
the  tube  so  as  to  cause  a  current  of  air  to  pass  through  it,  a  very 
loud  note  will  be  produced. 

315.  Vibrations   of  Rods   and    Laminae. — A    plate    of 
metal  called  a  reed  is  much  used  for  musical  purposes  in  connec- 
tion with  a  column  of  air,  as  already  stated.     In  parlor  organs 
the  sound  is  produced  by  the  action  of  vibrating  reeds  upon  air 
currents,  just  as  in  the  case  of  the  musical  toy  called  harmonicon. 
Except  in  such  connection,  the  sounds  of  wires  and  laminae  are 
generally  too  feeble  to  be  employed  in  music.     But  their  vibra- 
tions have  been  much  studied,  on  account  of  the  interesting  phe- 
nomena attending  them. 

Such  vibrations  afford  a  convenient  mode  of  determining  the 
velocity  of  sound  in  solids.  A  rod,  held  firmly  by  a  clamp  in  the 
middle,  and  rubbed  about  half  way  between  the  middle  and  the 
end  by  a  leather  pad  well  resined,  will  give  a  note  due  to  longi- 
tudinal vibrations  of  the  rod.  While  the  rod  gives  out  its  funda- 
mental note  the  ends  vibrate  freely,  being  neither  compressed  nor 
extended  :  but  at  the  centre,  held  by  the  clamp,  there  is  no  vibra- 
tion, but  a  maximum  effect  of  alternate  compression  when  the  two 
pulses  meet,  and  extension  when  they  again  travel  toward  the 
ends.  The  middle  of  the  rod  is  a  node.  The  time  of  a  complete 
vibration  is  the  time  that  a  pulse  would  require  to  travel  twice 
the  length  of  the  rod.  If  the  note  given  is  due  to  512  vibrations 
per  second,  and  the  length  of  the  rod  be  x  feet,  then  a  length  of 
%x  feet  is  passed  over  512  times  in  a  second,  and  the  velocity  in 
the  substance  of  the  rod  is  512  x  2#  feet  per  second. 

316.  Wires. — If  one  end  of  a  steel  wire  is  fastened  in  a  vise 
and  vibrated,  while  a  thin  blade  of  sunlight  falls  across  it,  the 
path  of  the  illuminated  point  may  be  traced.     It  is  not  ordinarily 
a  circular  arc  about  the  fixed  point  as  a  centre,  but  some  irregular 
figure  ;  and  frequently  the  point  describes  two  systems  of  ellipses, 
the  vibrations  passing  alternately  from  one  system  to  the  other 
several  times  before  running  down.     If  the  structure  of  the  wire 
were  the  same  in  every  part  across  its  section,  and  if  the  fastening 
pressed  equally  on  every  point  around  it,  the  orbit  of  each  particle 
would  be  a  series  of  ellipses,  whose  major  axes  are  on  the  same 
line.     If,  moreover,  there  was  no  obstruction  to  the  motion,  and 
the  law  of  elasticity  could  obtain  perfectly,  it  would  vibrate  in  the 
same   elliptic   orbit   forever,   the  force  toward   the  centre  being 
directly  as  the  distance.     It  is  easy  to  cause  the  wire,  in  the  ex- 
periment just  described,  to  vibrate  also  in  parts  ;  in  which  case 
each  atom,  while  describing  the  elliptic  orbit,  will  perform  several 


BELLS.  207 

smaller  circuits,  which  appear  as  waves  on  the  circumference  of  the 
larger  figure. 

317.  Chladni's  Plates. — If  a  square  plate  of  glass  or  elastic 
metal,  of  uniform  thickness  and  density,  be. fastened  by  its  centre 
in  a  horizontal  position,  and  a  bow  be  drawn  on  its  edge,  it  will 
emit  a  pure  musical  tone  ;  and  by  varying  the  action  of  the  bow, 
and  touching  different  points  of  the  edge  with  the  finger,  a  variety 
of  sounds  may  be  obtained  from  it.  The  plate  necessarily  vibrates 
in  parts ;  and  the  lowest  pitch  is  produced  when  there  are  two 
nodal  lines  parallel  to  the  sides,  and  crossing  at  the  centre,  thus 
dividing  the  plate  into  four  square  ventral  segments.  The  posi- 
tion of  the  nodal  lines,  and  the  forms  of  the  segments,  are  beauti- 
fully exhibited  by  sprinkling  writing-sand  on  the  plate.  The  par- 
ticles will  dance  about  rapidly  till  they  find  the  lines  of  rest,  where 
they  will  presently  be  collected.  For  every  new  tone  the  sand  will 
show  a  new  arrangement  of  nodal  lines ;  and  as  two  or  more 
modes  of  vibration  may  coexist  in  plates,  as  well  as  in  strings  and 
columns  of  air,  the  resultant  nodes  will  also  be  rendered  visible. 
Again,  by  fastening  the  plate  at  a  different  point,  still  other  ar- 
rangements will  take  place,  each  distinguishable  by  the  position 
of  its  nodal  lines  and  the  pitch  of  its  musical  note.  The  form  of 
the  plate  itself  may  also  be  varied,  and  each  form  will  be  charac- 
terized by  its  own  peculiar  systems.  Chladni,  who  first  performed 
these  interesting  experiments,  delineated  and  published  the  forms 
of  ninety  different  systems  of  vibration  in  the  square  plate  alone. 

If  a  fine  light  powder,  as  lycopodium  (the  pollen  of  a  species 
of  fern),  be  scattered  on  the  plate,  it  is  affected  in  a  very  different 
manner  from  he;:vy  sand.  It  will  gather  into  rounded  heaps  on 
those  portions  of  the  segments  which  have  the  greatest  amplitude 
of  vibration  ;  the  particles  which  compose  the  heaps  performing  a 
continual  circulation,  down  the  sides  of  the  heaps,  along  the  plate 
to  the  centre,  and  up  the  axis.  If  the  vibration  is  violent,  the 
heaps  will  be  thrown  up  frem  the  plate  in  little  clouds  over  the 
portions  of  greatest  motion.  The  cause  of  this  singular  effect  was 
ascertained  by  Faraday,  who  found  that  in  an  exhausted  receiver 
the  phenomenon  ceased.  It  is  due  to  a  circulation  of  the  air, 
which  lies  in  contact  with  a  vibrating  plate.  The  air  next  to 
those  parts  which  have  the  greatest  amplitude  is  at  each  vibration 
thrown  upward  more  powerfully  than  elsewhere,  and  surround- 
ing particles  press  into  its  place,  and  thus  a  circulation  is  estab- 
lished ;  and  a  fine  light  powder  is  more  controlled  by  these  at- 
mospheric movements  than  by  the  direct  action  of  the  plate. 

818.  Bells. — If  a  thin  plate  of  metal  takes  the  form  of  a  cylin- 
der or  bell,  its  fundamental  note  is  produced  when  each  ring  of  the 


•208  ACOUSTICS. 

material  changes  from  a  circle  to  an  ellipse,  and  then  into  a  second 
ellipse,  whose  axis  is  at  right  angles  to  that  of  the  former,  as  seen  in 
Fig.  199.  It  thus  has  four  ventral  seg- 

"Fir*     •«  QQ 

jnents  and  four  nodal  lines,  the  latter  lying 
in  the  plane  of  the  axis  of  the  bell  or  cylin- 
der. If  the  rings  which  compose  the  bell 
were  all  detached  from  one  another,  they 
would  have  different  rates  of  vibration  ac- 
•cording  to  their  diameter,  and  hence  would 
produce  tones  of  various  pitch  ;  but,  being 
bound  together  by  cohesion,  they  are  com- 
pelled to  keep  the  same  time,  and  hence 
give  but  one  fundamental  tone.  But  a 
bell,  especially  if  quite  thin,  may  be  made  to  emit  a  series  of  har- 
monic sounds  by  dividing  up  into  a  greater  number  of  segments. 
It  is  obvious  that  the  number  of  nodes  must  always  be  even,  be- 
cause two  successive  segments  must  move  in  opposite  directions 
in  one  and  the  same  instant ;  otherwise  the  point  between  them 
could  not  be  kept  at  rest,  and  therefore  would  not  be  a  node.  Be- 
sides the  principal  tone  of  a  church-bell,  one  or  two  subordinate 
•sounds  oh  a  different  pitch  may  usually  be  detected.  A  glass  bell, 
suitably  mounted  for  the  lecture-room,  will  yield  ten  or  twelve 
harmonics,  by  means  of  a  bow  drawn  on  its  edge. 

319.  The  Voice. — The  vocal  organ  is  complex,  consisting 
•of  a  cavity  called  the  larynx,  and  a  pair  of  membranous  folds  like 
valves,  having  a  narrow  opening  between  them;  this  opening, 
called  the  glottis,  admits  the  air  to  the  larynx  from  the  wind-pipe 
below.  The  edges  of  these  valves  are  thickened  into  a  sort  of 
cord,  and  for  this  reason  the  apparatus  is  called  the  vocal  cords.  In 
the  act  of  breathing,  the  folds  of  the  glottis  lie  relaxed  and  sepa- 
rate from  each  other,  and  the  air  passes  freely  between  them,  with- 
out producing  vibration.  But  in  the  effort  to  form  a  vocal  sound, 
they  approach  each  other,  and  become  tense,  so  that  the  current 
of  air  throws  them  into  vibration.  These  vibrations  are  enforced 
by  the  resonant  vibrations  of  the  air  of  the  larynx  above ;  and 
thus  a  fullness  of  sound  is  produced,  as  in  many 
musical  instruments,  in  which  a  reed,  and  the  air 
of  a  cavity,  perform  synchronous  vibrations,  and 
emit  a  much  louder  sound  than  either  could  do 
alone.  If  two  pieces  of  thin  india-rubber  be 
stretched  across  the  end  of  a  tube,  with  their 
edges  parallel,  and  separated  by  a  narrow  space, 
as  represented  in  Fig.  200,  the  arrangement  will 
.give  an  idea  of  the  larynx  and  glottis  of  the  vocal 


THE     EAR.  209 

organ.     If  air  be  forced  through,  a  sound  is  produced,  whose 
pitch  depends  on  the  size  of  the  tube  and  the  tension  of  the  valves. 

The  natural  key  of  a  person's  voice  depends  on  the  length  and 
weight  of  the  vocal  cords,  and  the  size  of  the  larynx.  The  yield- 
ing nature  of  all  the  parts,  and  the  ability,  by  muscular  action,  to" 
change  the  form  and  size  of  the  cavity  and  the  tension  of  the 
valves,  give  great  variety  to  the  pitch,  and  the  power  of  adjusting 
it  with  precision  to  every  shade  of  sound  within  certain  limits. 
No  instrument  of  human  contrivance  can  be  brought  into  com- 
parison with  the  organ  of  voice.  After  the  voice  is  formed  by  its 
appropriate  organ,  it  undergoes  various  modifications,  by  means 
of  the  palate,  the  tongue,  the  teeth,  the  lips,  and  the  nose,  before 
it  is  uttered  in  the  form  of  articulate  speech. 

320.  The  Organ  of  Hearing.— The  principal  parts  of  the 
ear  are  the  following : 

1.  The  outer  ear,  E  a  (Fig.  201),  terminating  at  the  mem- 
brane of  the  tympanum,  m. 

FIG.  201. 


2.  The  tympanum,  a  cavity  separated  from  the  outer  ear  by  a 
membrane,  m,  and  containing  a  series  of  four  very  small  bones 
(ossicles),  b,  c,  o   and  s,  severally  called,  on  account  of  their  form, 
the  hammer,  the  anvil,  the  ball,  and  the  stirrup.     The  figure 
represents  the  walls  of  the  tympanum  as  mostly  removed,  in  order 
to  show  the  internal  parts.    This  cavity  is  connected  with  the 
back  part  of  the  mouth  by  the  Eustachian  tube,  d. 

3.  The  labyrinth,  consisting  of  the  vestibule,  v,  the  semicircular 
canals,  f,  and  the  cochlea,  g.     The  latter  is  a  spiral  tube,  winding 
two  and  a  half  times  round.     The  parts  of  the  labyrinth  are  exca- 
vated in  the  hardest  bone  of  the  body.    The  figure  shows  only  its 
exterior.     There  are  two  orifices  through  the  bone  which  sepa- 


210  ACOUSTICS. 

rates  the  labyrinth  from  the  tympanum,  the  round  orifice,  e,  pass- 
ing into  the  cochlea,  and  the  oval  orifice,  s,  leading  to  the  vesti- 
bule. These  orifices  are  both  closed  by  a  thin  membrane.  The 
ossicles  of  the  tympanum  forma  chain  which  connects  the  centre 
of  the  membrane,  m,  with  that  which  closes  the  oval  orifice.  The 
labyrinth  is  filled  with  a  liquid,  in  various  parts  of  which  float 
the  fibres  of  the  auditory  nerve. 

By  the  form  of  the  outer  ear,  the  waves  are  concentrated  upon 
the  membrane  of  the  tympanum,  thence  conveyed  through  the 
chain  of  bones  to  the  membrane  of  the  labyrinth,  and  by  that  to 
the  liquid  within  it,  and  thus  to  the  auditory  nerve,  whose  fibres 
lie  in  the  liquid. 


CHAPTER    IV. 

MUSICAL  SCALES— THE  RELATIONS  OF  MUSICAL  SOUNDS. 

321.  Numerical  Relations  of  the  Notes.— To  obtain  the 
series  of  notes  which  compose  the  common  scale  of  music,  it  is 
convenient  to  use  the  monochord.  Calling  the  sound,  which  is 
given  by  the  whole  length  of  the  string,  the  fundamental,  or  key 
note,  of  the  scale,  we  measure  off  the  following  fractions  of  the 
whole  for  the  successive  notes,  namely  :  $•,  $,  f ,  f ,  f ,  ^,  |.  If  the 
whole,  and  these  fractions,  are  made  to  vibrate  in  order,  the  ear 
will  recognize  the  sounds  as  forming  the  series  called  the  gamuty 
or  diatonic  scale. 

Now  as  the  number  of  vibrations  varies  inversely  as  the  length 
of  the  string,  the  number  of  vibrations  of  these  notes  respectively, 
expressed  in  fractions  of  the  number  of  vibrations  of  the  whole 
string,  which  we  will  call  1,  will  be  1,  f ,  f ,  f .  f ,  |,  *$-,  2. 

If  the  whole  string  vibrates  128  times  per  second,  $  of  the 
string  would  give  |  of  128  vibrations,  or  144  vibrations ;  \  of  the 
string  would  give  £  of  128  vibrations  per  second,  or  1 60  vibrations. 
To  express  the  relative  number  of  vibrations  in  the  series  above, 
reduce  the  fractions  to  a  common  denominator  and  compare  their 
numerators,  and  we  have 

24,  27^30,032,^-36^  40,a45^  48. 

Sounds  whose  vibrations  per  second  bear  to  each  other  the 
ratios  of  the  series  above  are  not  arbitrarily  chosen  to  form  the 
scale,  but  they  are  demanded  by  the  ear.  The  notes  correspond- 
ing to  the  series  are  named  according  to  their  place  in  the  series  ; 
thus  a  note  whose  vibrations  are  ff  of  the  vibrations  of  the 
fundamental,  is  called  the  third,  one  whose  vibrations  are  -f-£  is 


THE     INTERVALS.  211 

the  fifth,  and  that  whose  vibrations  are  -ff,  or  twice  as  many  as 
those  of  the  fundamental,  is  the  eighth  or  octave. 

322.  Relations  of  the   Intervals.  —  An  interval  is  the 
relative  pitch  of  two  sounds,  and  its  numerical  value  is  expressed 
by  a  fraction  whose  numerator  is  the  number  of  vibrations  per 
second  of  the  higher  sound,  and  whose  denominator  is  the  num- 
ber of  vibrations  of  the  lower  or  graver  sound,  or  by  any  fraction 
equal  to  this. 

In  examining  the  relation  of  each  two  successive  numbers  in 
the  foregoing  series,  we  find  three  different  ratios.     Thus, 
27  :  24,  36  :  32,  and  45  :  40,  is  each  as    9  :    8. 

30  :  27 and  40  :  36, 10  :    9. 

32  :  30 and  48  :  45, 16  :  15. 

Therefore,  of  the  seven  successive  intervals,  in  the  diatonic 
scale,  there^  are  three  equal  to  |,  two  equal  Jff°-,  and  two  others 
equal  to  -|-f .  Each  of  the  first  five  is  called  a  tone ;  each  of  the 
last  two  is  called  a  semitone. 

323.  Repetition  of  the  Scale.— The  eighth  note  of  the 
scale  so  much  resembles  the  first  in  sound,  that  it  is  regarded  as  a 
repetition  of  it,  and  called  by  the  same  name.     Beginning,  there- 
fore, with  the  half  string,  where  the  former  series  closed,  let  us 
consider  the  sound  of  that  as  the  fundamental,  and  take  f  of  it 
for  the  second,  f  of  it  for  the  tbird,  &c.;  we  then  close  a  second 
series  of  notes  on  the  quarter-string,  whose  sound  is  also  con- 
sidered a  repetition  of  the  former  fundamental.     Each  fraction 
of  the  string  used  in  the  second  scale  is  obviously  half  of  the  cor- 
responding fraction  of  the  whole  string,    and  therefore  its  note  an 
octave  above  the  note  of  that.    This  proc'ess  may  be  repeated  in- 
definitely, giving  the  second  octave,   third  octave,  &c.    Ten  or 
eleven  octaves  comprehend  all  sounds  appreciable  by  the  human, 
ear ;  the  vibrations  of  the  extreme  notes  of  this  entire  range  have 
the  ratio  of  I  :  210,  or  1  :  211 ;  that  is,  1  :  1024,  or  1  :  2048.   Hence, 
if  16  vibrations  per  second  produce  the  lowest  appreciable  note, 
the  highest  varies  from  16,000  to  33,000.     It  was  ascertained  by 
Dr.  Wollaston  that  the  highest  limit  is  different  for  different  ears  ; 
so  that  when  one  person  complains  of  the  piercing  shrillness  of  a 
sound,  another  maintains  that  there  is  no  sound  at  all.      The 
lowest  limit  is  indefinite  for  a  different  reason ;  the  sounds  are 
heard  by  all,  but  some  will  recognize  them  as  low  musical  tones, 
while  others  only  perceive  a  rattling  or  fluttering  noise.    Few 
musical  instruments  comprehend  more  than  six  octaves,  and  the 
human  voice  has  only  from  one  to  three,  the  male  voice  being  in 
pitch  an  octave  lower  than  the  female. 

324.  Modes  of  Naming  the  Notes. — There  is  one  system 


212  ACOUSTICS. 

of  names  for  the  notes  of  the  scale,  which  is  fixed,  and  another 
which  is  movable.  The  first  is  by  the  seven  letters,  A,  B,  G,  D, 
E,  F,  G.  The  notes  of  the  second  octave  are  expressed  by  the  same 
letters,  in  some  way  distinguished  from  the  former.  The  best 
method  is  to  write  by  the  side  of  the  letter  the  numeral  expressing 
that  index  of  2,  which  corresponds  to  the  octave:  as  A^,  A3, 
&c.,  in  the  octaves  above  ;  AT,  A%,  in  those  below. 

Any  note  may  be  designated  as  C,  but  this  letter  is  usually 
assigned  to  that  note  which  is  due  to  256  vibrations  per  second, 
or  middle  C  of  the  piano-forte. 

The  second  mode  of  designation  is  by  the  syllables,  do,  re,  mi, 
fa,  sol,  la,  si.  These  express  merely  the  relations  of  notes  to  each 
other,  do  always  being  the  fundamental,  re  its  second,  mi  its  third, 
&c.  In  the  natural  scale,  do  is  on  the  letter  C,  re  on  D,  &c.;  but 
by  the  aid  of  interpolated  notes,  the  scale  of  syllables  may  be 
transferred,  so  as  to  begin  successively  with  every  letter  of  the 
fixed  scale. 

325.  The  Chromatic  Scale. — Let  the  notes  of  the  diatonic 
scale  be  represented  (Fig.  202)  by  the  horizontal  lines,  C,  D,  &c.; 
the  distance  from  C  to  D  being  a  tone,  from  D  to  E  a 
tone,  E  to  Fa,  semitone,  &c.     It  will  be  observed  that 
the  fundamental,  C,  is  so  situated  that  there  are  two 
whole  tones  above  it,  before  a  semitone  occurs,  and  then 
three  whole  tones  before  the  next  semitone.     C  is  there-       C2 

L LL          fore  the  letter  to  be  called  by  the  syllable  do,  in  order  to 

bring  the  first  semitone  between  the  3d  and  4th,  and 
the  other  semitone  between  the  7th  and  8th,  as  the 
figure  represents  them.  Now,  that  we  may  be  able  to 
transfer  the  scale  of  relations  to  every  part  of.  the  fixed 
scale  (which  is  necessary,  in  order  to  vary  the  character 
of  music,  without  throwing  it  beyond  the  reach  of  the 
voice),  the  whole  tones  are  bisected,  and  two  semitone 
intervals  occupy  the  place  of  each.  The  dotted  lines 
in  the  figure  show  the  places  of  the  interpolated  notes, 
which,  with  the  original  notes  of  the  diatonic  scale,  di- 
vide the  whole  into  a  series  of  semitones.  This  is  called 
the  chromatic  scale.  The  interpolated  note  between  C 
and  D  is  written  G  #  (G  sharp),  or  D  b  (D  flat),  and  so 
of  the  others.  As  the  whole  tones  lie  in  groups  of  twos 
and  threes,  so  the  new  notes  inserted  are  grouped  in  the 
same  way.  This  explains  the  arrangement  of  the  black 
keys  by  twos  and  threes  alternately  in  the  key-board  of  the  organ 
and  piano-forte.  The  white  keys  compose  the  diatonic  scale,  the 
white  and  black  keys  together,  the  chromatic  scale.  It  is  obvious 


CHORDS     AND    DISCORDS.  213 

that  on  the  chromatic  scale  any  one  of  the  twelve  notes  which 
compose  it  may  become  do,  or  the  fundamental  note,  since  the  re- 
quired series,  2  tones,  1  semitone,  3  toues,  1  semitone,  can  be 
arranged  to  succeed  each  other,  at  whatever  note  we  begin  the 
reckoning.  This  change,  by  which  the  fundamental  note  is  made 
to  fall  on  different  letters,  is  called  the  transposition  of  the  scale. 

326.  Chords  and  Discords. — When  two  or  more  sounds, 
meeting  the  ear  at  once,  form  a  combination  which  is  agreeable, 
it  is  called  a  chord ;  if  disagreeable,  a  discord.     The  disagreeable 
quality  of  a  discord,  if  attended  to,  will  be  perceived  to  consist  in 
a  certain  roughness  or  harshness,  however  smooth  and  pure  the 
simple  sounds  which  are  combined.     On  examining  the  combina- 
tions, it  will  be  found  that  if  the  vibrations  of  two  sounds  are  in 
some  very  simple  relations,  as  1  :  2, 1  :  3,  2  :  3,  3  :  4,  &c.,  they  pro- 
duce a  chord ;  and  the  lower  the  terms  of  the  ratio,  the  more  per- 
fect the  chord.     On  the  other  hand,  if  the  numbers  necessary  to 
express  the  relations  of  the  sounds  are  large,  as  8  :  9,  or  15  :  16,  a 
discord  is  produced.     It  appears  that  concordant  sounds  have  fre- 
quent coincidences  of  vibrations.    If,  in  two  sounds,  there  is  coin- 
cidence at  every  vibration  of  each,  then  the  pitch  is  the  same,  and 
the  combination  is  called  unison.     If  every  vibration  of  one  coin- 
cides with  every  alternate  vibration  of  the  other,  the  ratio  is  1  :  2, 
and  the  chord  is  th^  octave,  the  most  perfect  possible.    The  fifth  is 
the  next  most  perfect  chord,  where  every  second  vibration  of  the 
lower  meets  every  third  of  the  higher,  2  :  3.    The  fourth,  3  :  4,  the 
major  third,  4  :  5,  the  minor  third,  5  :  6,  and  the  sixth,  3  :  5,  are 
reckoned  among  chords  ;  while  the  second,  8  :  9,  and  the  seventh, 
S  :  15,  are  harsh  discords.     What  is  called  the  common  chord  con- 
sists of  the  1st,  3d,  and  5th,  combined,  and  is  far  more  used  in 
music  than  any  other.    Harmony  consists  of  a  succession  of  chords, 
or  rather,  of  such  a  succession  of  combined  sounds  as  is  pleasing 
to  the  ear  ;  for  discords  are  employed  in  musical  composition, 
their  use  being  limited  by  special  rules.     Many  combinations, 
which  would  be  too  disagreeable  for  the  ear  to  dwell  upon,  or  to 
finish  a  musical  period,  are  yet  quite  necessary  to  produce  the  best 
effect ;  and  without  the  relief  which  they  give,  perfect  harmony, 
if  long  continued,  would  satiate. 

327.  Temperament. — This  is  a  term  applied  to  the  small 
errors  introduced  into  the  notes,  in  tuning  an  instrument  of  fixed 
keys,  in  order  to  adapt  the  notes  equally  to  the  several  scales.     If 
the  tones  were  all  equal,  and  if  semitones  were  truly  half  tones, 
no  such  adjustment  of  notes  would  be  needed ;  they  would  all 
be  exactly  correct  for  every  scale.     Representing  the  notes   in 


214  ACOUSTICS. 

the  scale  whose  fundamental  is  C  by  the  numbers  in  Art.  321,  we 
have, 

C,    D,    E,   F,    G,   A,   B  Cz,  Dz,  Ez,  &c. 

24,  27,  30,  32,  36,  40,  45,  48,   54,    60,   &c. 

Now  suppose  we  wish  to  make  D,  instead  of  C,  our  key-note  ; 
then  it  is  obvious  that  E  will  not  be  exactly  correct  for  tn"e  second 
on  the  new  scale.  For  the  fundamental  to  its  second  is  as  8  to  9  ; 
and  8  :  9  ::  27  :  30.375,  instead  of  30.  Therefore,  if  D  is  the  key- 
note, we  must  have  a  new  E,  slightly  above  the  E  of  the  original 
scale.  So  we  find  that  A,  represented  by  40,  will  not  serve  to  b& 
the  5th  in  the  new  scale;  since  2  :  3  ::  27  :  40.5,  which  is  a  little 
higher  than  A  (=  40).  After  adding  these  and  other  new  notes, 
to  render  the  intervals  all  exactly  right  for  the  new  key  of  D,  if 
we  proceed  in  the  same  manner,  and  ma"ke  J2(=  30)  our  key-note, 
and  obtain  its  second,  third,  &c.,  exactly,  we  shall  find  some  of 
them  differing  a  little,  both  from  those  of  the  key  of  C,  and  also 
of  the  key  of  D.  Using,  in  this  way  all  the  twelve  notes  of  the 
chromatic  scale  in  succession  for  the  fundamental,  it  appears  that 
several  different  E's,  F's,  G's,  &c.,  are  required,  in  order  to  make 
each  scale  perfect.  In  instruments,  whose  sounds  cannot  be  mod- 
ified by  the  performer,  like  the  organ  and  piano-forte,  as  it  is  con- 
sidered impossible  to  insert  all  the  pipes  or  strings  necessary  to 
render  every  scale  perfect,  such  an  adjustment  is  made  as  to  dis- 
tribute these  errors  equally  among  all  the  scales.  For  example,  E 
is  not  made  a  perfect  third  for  the  key  of  C,  lest  it  should  be  too 
imperfect  for  a  second  in  the  key  of  D,  and  for  its  appropriate 
place  in  other  scales.  It  is  this  equal  distribution  of  errors  among 
the  several  scales  which  is  called  temperament.  The  errors,  when 
thus  distributed,  are  too  small  to  be  observed  by  most  persons ; 
whereas,  if  an  instrument  was  tuned  perfectly  for  any  one  scale, 
all  others  would  be  intolerable. 

The  word  temperament,  as  above  explained,  has  no  application 
except  to  instruments  of  fixed  keys,  as  the  organ  and  piano-forte ; 
for,  where  the  performer  can  control  and  modify  the  notes  as  he  is 
playing,  he  can  make  every  key  perfect,  and  then  there  are  no 
errors  to  be  distributed.  The  flute-player  can  roll  the  flute 
slightly,  and  thus  humor  the  sound,  so  as  to  cause  the  same 
fingering  to  give  a  precisely  correct  second  for  one  scale,  a  correct 
third  for  another,  and  so  on.  The  player  on  the  violin  does  the 
same,  by  touching  the  string  in  points  slightly  different.  The  or- 
gans of  the  voice,  especially,  can  be  adjusted  to  make  the  intervals 
perfect  on  every  scale.  In  these  cases  there  is  no  tempering,  or 
dividing  of  errors  among  different  scales,  but  a  perfect  adjustment 
to  each  scale,  by  which  all  error  is  avoided. 


TIMBRE.  215 

328.  Harmonics. — The  fact  has  been  mentioned  that  a  string, 
or  a  column  of  air,  may  vibrate  in  parts,  even  while  vibrating  as 
a  whole.      It  only  remains  to  show  the  musical  relations  of  the 
sounds  thus  produced.     When  a  string  vibrates  in  parts,  it  divides 
into  halves,  thirds,  fourths,  or  other  aliquot  parts.    Now,  a  half- 
string  produces  an  octave  above  the  whole,  making  the  most  per- 
fect chord  with  it.     The  third  of  a  string  being  two-thirds  of  the 
half-string,  produces  the  fifth  above  the  octave,  a  very  perfect 
chord.     The  quarter-string  gives  the  second  octave  ;  the  fifth  part 
of  it,  being  f  of  the  quarter,  gives  the  major  third  above  the 
second  octave  ;  and  the  sixth  part,  being  f  of  the  quarter,  gives 
the  fifth  above  the  second  octave.    Thus,  all  the  simpler  divisions, 
which  are  the  ones  most  likely  to  occur,  are  such  as  produce  the 
best  chords ;  and  it  is  for  this  reason  that  the  sounds  are  called 
harmonics.     The  same  is  true  of  air-columns   and  bells.     The 
JEolian.  harp  furnishes  a  beautiful  example  of  the  harmonics  of  a 
string.     Two  or  more  fine  smooth  cords  are  fastened  upon  a  box, 
and  tuned,  at  suitable  intervals,  like  the  strings  of  a  violin;  and 
the  box  is  placed  in  a  narrow  opening,  where  a  current  of  air 
passes.     Each  string  at  different  times,  according  to  the  intensity 
of  the  breeze,  will  emit  a  pure  musical  note ;  and,  with  every 
change,  will  divide  itself  in  a  new  mode,  and  give  another  pitch, 
while  it  will  frequently  happen  that  the  vibrations  of  different 
divisions  will  coexist,  and  their  harmonic  sounds  mingle  with  each 
other. 

329.  Overtones. — But  the  parts  into  which  a  sounding  body 
divides  do  not  always  harmonize  with  the  whole.    For  instance,  ^ 
or  -^  of  a  string  is  discordant  with  the  fundamental.     The  word 
harmonics  is  not,  therefore,  applicable  except  to  a  very  few  of  the 
many  possible  sounds  which  a  body  may  produce.     Tne  word 
overtone  is  used  to  express  in  general  any  sound  whatever,  given 
by  a  part  of  a  sounding  body.     A  string  may  furnish  20  or  30 
overtones,  but  only  a  small  number  of  them  would  be  harmonics. 

The  presence  of  these  overtones  may  be  determined  by 
means  of  the  resonator  devised  by  Helmholtz  and  modified  by 
Konig,  which  can  be  adjusted  to  respond  to  a  great  variety  of 
notes  ;  if  on  drawing  out  the  cylinder  a  tone  is  produced,  it  must 
be  that  the  same  tone  exists  in  the  compound  sound  under  inves- 
tigation. 

330.  Timbre,  or  Quality  of  Tone.— Even  when  the  pitch 
of  two  sounding  bodies  is  the  same,  the  ear  almost  always  distin- 
guishes one  sound  from  the  other  by  certain  qualities  of  tone 
peculiar  to  each.     Thus,  if  the  same  letter  be  sounded  by  a  flute 


216  ACOUSTICS. 

and  the  string  of  a  piano,  each  note  is  easily  distinguished  from 
the  other.  Two  church-bells  may  be  upon  the  same  key,  and  yet 
one  be  agreeable,  and  the  other  harsh  to  the  ear. 

As  a  result  of  his  researches  Helmholtz  decides  that  the  timbre 
is  determined  by  the  overtones  which  accompany  the  primary 
tones.  If  these  overtones  could  be  eliminated,  leaving  only  the 
pure,  simple  tones,  a  note  of  given  pitch  sounded  by  a  flute  would 
not  be  distinguishable  from  a  note  of  the  same  pitch  sounded  by 
a  violin. 

A  long  monochord  can,  by  varying  the  mode  of  exciting  the 
vibrations,  be  made  to  yield  a  great  variety  of  sounds,  while  there 
is  perceived  in  them  all  the  same  fundamental  undertone  which 
determines  the  pitch.  If  the  string  be  struck  at  the  middle,  then 
no  node  can  be  formed  at  that  point ;  hence,  the  mixed  sound  will 
contain  no  overtones  of  the  -J-,  £,  f,  £,  TV,  or  other  even  aliquot 
parts  of  the  string ;  for  all  such  would  require  a  node  at  the 
middle.  But  if  struck  at  one-third  of  its  length  from  the  endr 
then  the  overtones,  £,  £,  &c.,  may  exist,  but  not  those  of  J,  ^,  -J> 
or  any  other  parts  whose  node  would  fall  at  £  of  the  length  from 
the  end. 

For  reasons  which  are  mostly  unknown,  some  sounding  bodies 
have  their  fundamental  accompanied  by  harmonic  overtones,  and 
others  by  overtones  which  are  discordant.  And  this  is  one  cause 
of  the  agreeable,  or  unpleasant,  quality  of  the  sounds  of  different 
bodies. 

331.  Communication  of  Vibrations. — The  acoustic  vibra- 
tions of  one  body  are  readily  communicated  to  others,  which  are 
near  or  in  contact.  We  have  already  noticed  that  the  vibrations 
of  a  reed  will  excite  those  of  a  column  of  air  in  a  pipe.  If  two 
strings,  which  are  adapted  to  vibrate  alike,  are  fastened  on  the 
same  box,  and  one  of  them  is  made  to  sound,  the  other  will  sound 
also  more  or  less  loudly,  according  to  the  intimacy  of  their  connec- 
tion. The  vibrations  are  communicated  partly  through  the  air, 
and  partly  through  the  materials  of  the  box.  So,  if  a  loud  sound 
is  uttered  near  a  piano-forte,  several  strings  will  be  thrown  into 
vibration,  whose  notes  are  heard  after  the  voice  ceases.  The  no- 
ticeable fact  in  all  such  experiments  is,  that  the  vibrations  thus 
communicated  from  one  body  to  another  cause  sounds  which  har- 
monize with  each  other,  and  with  the  original  sound.  For  the- 
rate  of  vibration  will  either  be  identical,  or  have  those  simple  re- 
lations which  are  expressed  by  the  smallest  numbers.  Let  a  per- 
son hold  a  pneumatic  receiver  or  a  large  tumbler  before  him,  and 
utter  at  the  mouth  of  it  several  sounds  of  different  pitch  ;  and  he 
will  probably  find  some  one  pitch  which  will  be  distinctly  rein- 


CRISPATIONS     OF     FLUIDS.  217 

forced  by  the  vessel.  That  particular  note,  which  the  receiver  by 
its  size  and  form  is  adapted  to  produce,  will  not  be  called  forth  by 
a  sound  that  would  be  discordant  with  it.  The  melodeon,  ser- 
aphine,  and  instruments  of  like  character,  owe  their  full  and  bril- 
liant notes  to  reeds,  each  of  which  has  its  cavity  of  air  adapted  to 
vibrate  in  unison  with  it.  It  sometimes  happens  that  the  second 
body,  vibrating  as  a  whole,  would  not  harmonize  with  the  first,  and 
yet  Avill  give  the  same  note  by  some  mode  of  division.  Thus  it  is 
that  all  the  various  sounds  of  the  monochord,  and  of  the  strings 
of  the  viol,  arc  reinforced  by  the  case  of  thin  wood  upon  which 
they  are  stretched.  The  plates  of  wood  divide  by  nodal  lines  into 
some  new  arrangement  of  ventral  segments  for  every  new  sound 
emitted  by  the  string.  In  like  manner,  the  pitch  of  the  tuning- 
fork,  and  all  the  rapid  notes  of  a  music-box,  are  rendered  loud  and 
full  by  the  table,  in  contact  with  which  they  are  brought.  The 
extended  material  of  the  table  is  capable  .of  division  into  a  great 
variety  of  forms,  and  will  always  give  a  sound  in  unison  with  the 
instrument  which  touches  it. 

382.  One  System  of  Vibrations  Controlling  Another. — 

If  two  sounding  bodies  are  nearly,  but  not  precisely  on  the  same 
key,  they  will  sometimes,  when  brought  into  close  contact,  be 
made  to  harmonize  perfectly.  The  vibrations  of  the  more  power- 
ful will  be  communicated  to  the  other,  and  control  its  movements 
so  that  the  discordance,  which  they  produce  when  a  few  inches 
apart,  will  cease,  and  concord  will  ensue.  Two  diapason  pipes  of 
an  organ,  tuned  a  quarter-tone  or  even  a  semitone  from  unison,  so 
as  to  jar  disagreeably  upon  the  ear,  when  one  inch  or  more  asun- 
der, will  be  in  perfect  unison,  if  they  are  in  contact  through  their 
whole  length.  Even  the  slow  oscillations  of  two  watches  will  in- 
fluence each  other ;  if  one  gains  on  the  other  only  a  few  beats 
in  an  hour,  then,  if  they  are  placed  side  by  side  on  the  same  board, 
they  will  beat  precisely  together. 

333.  Crispations  of  Fluids. — Among  the  numerous  acous- 
tic experiments  illustrating  the  communication  of  vibrations,  none 
are  more  beautiful  than  those  in  which  the  vibrations  of  glass  rods 
are  conveyed  to  the  surface  of  a  fluid.  Let  a  very  shallow  pan  of 
glass  or  metal  be  attached  to  the  middle  of  a  thin  bar  of  wood, 
three  or  four  feet  long,  and  resting  near  its  ends  on  two  fixed 
bridges ;  let  water  be  placed  in  the  pan,  and  a  long  glass  rod 
standing  in  it,  or  on  the  wood,  be  vibrated  longitudinally,  by 
drawing  the  moistened  fingers  down  upon  it ;  the  liquid  immedi- 
ately shows  that  the  vibrations  are  communicated  to  it.  The  sur- 
face is  covered  with  a  regular  arrangement  of  heaps,  called  crixpa- 
tions,  which  vary  in  size  with  the  pitch  of  sound,  which  is  produced 


218  ACOUSTICS. 

by  the  same  vibration.  If  the  pitch  is  higher,  they  are  smaller, 
and  may  be  readily  varied  from  three  or  four  inches  in  diameter 
to  the  fineness  of  the  teeth  of  a  file.  Crispations  of  the  same 
character  are  also  formed  in  clusters  on  the  water  in  a  large  tum- 
bler or  glass  receiver,  when  the  finger  is  drawn  along  its  edge  ; 
every  ventral  segment  of  the  glass  produces  a  group  of  hillocks  by 
the  side  of  it  on  the  surface  of  the  water. 

334.  Interference  of  Waves  of  Sound.— Whenever  two 
sounds  are  moving  through  the  air,  every  particle  will,  at  a  given 
instant,  have  a  motion  which  is  the  resultant  of  the  two  motions 
which  it  would  have  had  if  the  sounds  were  separate.  These  mo- 
tions may  conspire,  or  they  may  oppose  each  other.  The  word 
interference  is  used  in  scientific  language  to  express  the  resultant 
effect,  whatever  it  may  be. 

If  two  sound  waves  of  equal  intensity  and  of  the  same  length 
move  together  so  that  any  phase  of  one  is  coincident  with  the  like 
phase  of  the  other  the  resultant  sound  has  greater  intensity  than 
either  of  the  components  ;  if,  however,  any  phase  of  one  is  coinci- 
dent with  the  opposite  phase  of  the  other,  entire  extinction  of  the 
sound  results,  and  we  have  silence.  To  illustrate  this  experi- 
mentally, take  two  pieces  of  tubing  and  bend  them  into  the  form 
shown  in  Fig.  203,  the  branch  A  being  large  enough  to  slide  over 

the  other,  and  insert  a 

FlG-  m  whistle  at  D.    The  sound 

waves  travel  to  the  ear 
placed  at  C  by  two  differ- 
ent routes,  starting  in 
the  same  phase  at  the 
point  D.  If  the  branches 
A  and  B  are  of  equal 

length,  or  differ  by  one  or  more  whole  wave  lengths,  the  waves 
will  meet  at  C  in  the  same  phase  and  produce  a  sound  of  greater 
intensity  than  that  of  either  alone ;  but  if  the  branch  A  be 
drawn  out  or  pushed  in  till  the  route  A  differs  in  length  from 
B  by  half  a  wave  length,  or  by  any  odd  multiple  of  half  a 
wave  length,  opposite  phases  will  meet  at  C  and  destroy  each 
other,  producing  silence.  A  much  more  simple  experiment  to 
show  the  same  effect  is  the  following :  The  two  prongs  of  a  tuning- 
fork  always  vibrate  in  opposite  directions,  one  producing  a  con- 
densation in  the  direction  in  which  the  other  produces  rarefaction, 
thus  destroying  each  other's  effect  by  interference,  and  hence  the 
almost  total  absence  of  sound  when  the  fork  is  held  free  in  the 
hand.  In  such  case,  if  the  sound  waves  from  one  prong  be  inter- 
cepted by  slipping  over  it  without  contact  a  paper  cylinder,  the 


LENGTH     OF    WAVE.  219 

sound  is  augmented.  Hold  a  vibrating  fork  so  that  either  the 
back  face  of  a  prong,  or  the  side  faces  of  the  two  prongs,  are 
parallel  to  the  ear,  and  the  sound  will  be  distinctly  audible  ;  turn 
the  fork  about  its  axis  45°  from  either  of  these  positions,  and 
silence  results.  During  one  entire  rotation  of  the  fork  there  will 
be  four  positions  of  maximum  intensity  and  four  other  positions 
of  total  extinction  of  the  sound.  If  the  fork  be  rotated  over  the 
mouth  of  a  resonating  jar,  the  effect  is  much  more  striking. 

The  beats,  which  are  frequently  heard  in  listening  to  two 
sounds,  indicate  the  points  of  maximum  condensation  produced 
by  the  union  of  the  condensed  parts  of  both  systems  of  waves. 
And  the  sounds  are  considered  discordant  when  these  beats  are 
just  so  frequent  as  to  produce  a  disagreeable  fluttering  or  rattling. 
If  too  near  or  too  far  apart  for  this,  they  are  regarded  practically 
as  concordant.  And  when  the  beats  are  too  close  to  be  perceived 
separately,  yet  the  peculiar  adjustment  of  condensations  of  one 
system  with  those  of  the  other,  according  as  one  wave  measures 
two,  or  two  waves  measure  three,  or  four  measure  Jive,  &c.,  is  at 
once  distinguished  by  the  ear,  and  recognized  as  the  chord  of  the 
octave,  the  fifth,  the  third,  &c.  When  a  sound  and  its  octave  are 
advancing  together,  there  are  instants  in  which  any  given  particle 
of  air  is  impressed  with  two  opposite  motions,  and  other  alternate 
moments  when  both  motions  are  in  the  same  direction.  For  the 
waves  of  the  highest  sound  are  half  as  long  as  those  of  the  lowest ; 
hence,  while  every  second  condensation  of  the  former  coincides 
with  every  condensation  of  the  latter,  the  alternate  ones  of  the 
former  must  be  at  the  points  of  greatest  rarefaction  of  the  latter ; 
and  this  cannot  occur  without  opposite  movements  of  the  par- 
ticles. 

335.  Number  and  Length  of  Waves  for  Each  Note.— 

Though  the  vibrations  of  any  musical  note  are  too  rapid  to  be 
counted,  yet  the  number  may  be  ascertained  in  several  ways. 

If  an  elastic  slip  of  metal  be  clamped  at  one  end  so  that  the 
other  end  may  rest  against  a  toothed  wheel,  and  the  wheel  be  re- 
volved with  different  velocities,  musical  notes  of  different  pitch 
will  be  produced.  If  a  uniform  velocity  be  maintained  for  a  given 
time,  sixty  seconds  for  instance,  and  the  number  of  revolutions  of 
the  toothed  wheel  be  read  from  an  indicator  suitably  connected 
with  the  wheel,  then  the  number  of  teeth  upon  the  wheel  multi- 
plied by  the  number  of  revolutions  gives  the  number  of  impulses 
communicated  to  the  air  in  the  given  time  ;  this  product,  divided 
by  the  seconds  in  the  time,  sixty  in  the  case  supposed,  gives  the 
number  of  vibrations  per  second.  The  instrument  is  called 
Savart's  Wheel. 


220  ACOUSTICS. 

Another  method  of  determining  the  number  of  vibrations  is  by 
means  of  the  siren,  invented  by  De  La  Tour.  In  this  instrument 
the  pulses  are  produced  by  puffs  of  air,  in  rapid  succession,  caused 
by  revolving  a  disc,  perforated  around  its  circumference  by 
numerous  holes  which  pass  in  front  of  an  air  jet.  The  calcu- 
lation is  similar  to  that  given  for  Savart's  wheel. 

In  the  method  given  above  it  is  difficult  in  practice  to  main- 
tain a  constant  velocity.  A  graphic  representation  of  the  vibra- 
tions, devised  by  Duhamel,  is  without  this  objection.  Without 
giving  details  of  the  construction,  the  principle  of  the  method 
may  be  given  as  follows : 

Support  the  vibrating  rod — a  tuning-fork  for  example — above 
a  table,  as  in  Fig.  204 ;  to  one  prong  of  the  fork  attach  a  fine  steel 
point  or  style,  which 

shall      very      lightly  FlG-  204- 

touch  the  strip  of 
paper  D  C,  which  has 
been  coated  with  a 
film  of  lamp  black ; 
place  at  /  an  electric 
style,  which  is  in  con- 
nection with  a  pendulum  beating  seconds,  which  shall  make  a 
dot  upon  the  paper  at  each  beat  of  the  pendulum.  If  the  paper 
be  moved  in  the  direction  D  C  while  the  fork  is  kept  vibrating, 
an  undulating  line  will  be  traced  upon  the  film,  and  the  number 
of  undulations  between  any  two  consecutive  second's  dots,  as  b  a, 
gives  the  number  of  vibrations  of  the  fork  per  second. 

In  these  ways  it  is  ascertained  that  the  numbers  corresponding 
to  the  letters  of  the  scale  are  the  following: 

(    (7,     D,    E,      F,      G,      A,      B,     C,, 
I  128,  144,  160,  170f,  192,  213$,  240,  256, 

The  highest  note  of  the  above  series,  (72,  256,  is  the  middle  G  of 
the  piano-forte. 

There  is  not,  however,  a  perfect  agreement  of  pitch  in  differ- 
ent countries,  and  among  different  classes  of  musicians.  Accord- 
ingly, C,  which  is  given  above  as  corresponding  to  128  vibrations 
per  second,  has  several  values,  varying  from  127  to  131. 

336.  Doppler's  Principle. — Thus  far  the  hearer  and  the 
source  of  sound  have  been  supposed  not  to  change  their  relative 
positions.  When  a  sounding  body  approaches  the  ear,  the  tone 
perceived  is  higher  than  that  due  to  the  number  of  vibrations  per 
second,  since  more  vibrations  per  second  reach  the  ear  than  if  the 
body  had  remained  at  rest  with  respect  to  the  hearer.  Suppose 
the  sound  to  be  middle  C,  and  the  sounding  body  and  the  hearer 


THE     PHONOGRAPH.  221 

to  remain  relatively  stationary,  then  256  vibrations  per  second 
will  be  communicated  to  the  ear  :  If  now  the  sounding  body  ap- 
proach at  the  rate  of  66  feet  per  second  there  will  be  perceived,  in 

(\fi 
addition    to  the  256  vibrations,  j-j  =  15  vibrations  per  second,. 

and  the  pitch  will  be  that  due  to  271  vibrations  per  second  in- 
stead of  256.  If  the  sounding  body  recede  from  the  hearer  the 
opposite  effect  will  be  produced. 

337.  Acoustic  Vibrations  Visibly  Projected.  —  The  vi- 
brations of  heavy  tuning-forks  can  be  magnified  and  rendered  dis- 
tinctly visible  to  an  audience  by  projecting  them  on  a  screen. 
The  fork  being  constructed  with  a  small  metallic  mirror  attached 
near  the  end  of  one  prong,  a  sunbeam  reflected  from  the  mirror 
will  exhibit  all  the  movements  of  the  fork  greatly  enlarged  on  a 
distant  wall ;  and  if  the  fork  is  turned  on  its  axis,  the  luminous 
projection  will  take  the  form  of  a  waving  line.     And  by  the  use  of 
two  forks,  all  the  phenomena  of  interference  may  be  rendered  as- 
distinct  to  the  eye  as  they  are  to  the  ear. 

338.  Edison's  Phonograph. — This  is  an  instrument  for  re- 
cording sounds  and  for  reproducing  them  at  any  subsequent  time. 
In  the  modern  instrument  the  records  are  taken  upon  cylinders  of 
wax.    A  cylinder  is  rotated  around  its  axis  at  a  uniform  speed.    At 
every  revolution  the  axis  is  displaced  in  its  own  direction  by  a  dis- 
tance equal  to  the  distance  between  two  contiguous  threads  of  a 
screw,  this  screw  being  cut  upon  one  end  of  the  shaft  upon  which 
the  cylinder  turns.     The  sound  to  be  recorded  is  received  upon  a 
diaphragm,  whose  plane  is  parallel  to  the  axis  of  the  cylinder.    The 
sound  makes  the  diaphragm  vibrate,  and  these  vibrations  cause  a 
cutting  style,  which  is  attached  to  the  rear  of  the  diaphragm,  to 
cut  in  the  wax  a  spiral  undulating  record  of  the  sound.     To  repro- 
duce the  sound  the  cutting  style  is  replaced  by  a  smooth,  pointed 
style.     This  is  placed  at  the  beginning  of  the  spiral  record,  and 
the  cylinder  is  again  revolved  at  its  former  speed.     The  style,  fol- 
lowing the  undulations,  causes  the  diaphragm  to  vibrate  in   the 
same  manner  as  when  it  was  subjected  to  the  original  sound  vibra- 
tions.    An  ear  placed  over  the  diaphragm  also  receives  these  as  if 
they  were  coming  from  the  original  source. 

The  phonograph  is  capable  of  recording  the  three  characteristics 
of  tones — pitch,  loudness,  and  quality.  The  nearer  together  the  in- 
dentations are,  the  higher  the  pitch  ;  the  deeper  they  are,  the  greater 
the  intensity ;  and  the  more  irregular  the  outline  of  the  individual 
indentations,  the  richer  in  overtones  ia  the  sound  recorded. 


PART    Y. 

OPTICS. 

CHAPTER    I. 

MOTION  AND  INTENSITY   OF  LIGHT. 

339.  Definitions. — Light  is  supposed  to  consist  of  exceed- 
ingly minute  and  rapid  vibrations  in  a  medium  or  ether  which 
fills  space ;  which  vibrations,  on  reaching  the  retina  of  the  eye, 
cause  vision,  as  the  vibrations  of  the  air  cause  hearing,  when  they 
impinge  on  the  tympanum  of  the  ear,  and  as  thermal  vibrations 
produce  a  sensation  of  warmth,  when  they  fall  on  the  skin  ;  the 
difference  between  light  and  heat  is  solely  a  difference  in  wave 
length,  waves  longer  than  those  of  the  extreme  red,  or  shorter 
than  the    extreme  violet,  producing  no  effect  upon   the  optic 
nerve. 

Bodies,  which  of  themselves  are  able  to  produce  vibrations  in 
the  ether  surrounding  them,  are  said  to  emit  light,  and  are  called 
self-luminous,  or  simply  luminous  ;  those,  which  only  reflect  light, 
are  called  non-luminous.  Most  bodies  are  of  the  latter  class.  A 
ray  of  light  is  a  line,  along  which  light  is  propagated  ;  a  beam  is 
made  up  of  many  parallel  rays ;  &  pencil  is  composed  of  rays  either 
diverging  or  converging  ;  and  is  not  unfrequently  applied  to  those 
which  are  parallel. 

A  substance,  through  which  light  is  transmitted,  is  called  a 
medium  ;  if  objects  are  clearly  seen  through  the  medium,  it  is 
called  transparent;  if  seen  faintly,  semi-transparent ;  if  light  is 
discerned  through  a  medium,  but  not  the  objects  from  which  it 
comes,  it  is  called  translucent ;  substances  which  transmit  no  light 
are  called  opaque,  though  all  are  more  or  less  translucent  when 
cut  in  sufficiently  thin  laminae. 

340.  Light  Moves  in  Straight  Lines. — So  long  as  the 
medium  continues  uniform,  the  line  of  each  ray  is  perfectly 


THE     VELOCITY     OF     LIGHT.  223 

straight.  For  an  object  cannot  be  seen  through  a  bent  tube ; 
and  if  three  discs  have  each  a  small  aperture  through  it,  a  ray 
cannot  pass  through  the  three,  except  when  they  are  exactly  in  a 
straight  line.  The  shadow  which  is  projected  through  space 
from  an  opaque  body  proves  the  same  thing;  for  the  edges  of  the 
shadow,  taken  in  the  direction  of  the  rays,  are  all  straight  Hues. 

From  every  point  of  a  luminous  surface  light  emanates  in  all 
possible  directions,  when  not  prevented  by  the  interposition  of 
an  opaque  body.  Thus,  a  candle  is  seen  by  night  at  the  distance 
of  one  or  two  miles ;  and  within  that  limit,  no  space  so  small  as 
the  pupil  of  the  eye  is  destitute  of  rays  from  the  candle.  A 
point  from  which  light  emanates  is  called  a  radiant.  If  light 
from  a  radiant  falls  perpendicularly  on  a  circular  disk,  the  pencil 
is  a  cone;  if  on  a  square  disk,  it  is  a  square  pyramid,  &c.,  the 
illuminated  surface  in  each  case  being  the  base,  and  the  radiant 
the  vertex. 

341.  The  Velocity  of  Light. — It  has  been  ascertained  by 
several  independent  methods,  that  light  moves  at  the  rate  of  about 
186,300  miles  per  second. 

One  method  is  by  means  of  the  eclipses  of  Jupiter's  satellites. 
The  planet  Jupiter  is  attended  by  four  moons  which  revolve  about 
it  in  short  periods.  These  small  bodies  are  observed,  by  the  tele- 
scope, to  undergo  frequent  eclipses  by  falling  into  the  shadow 
which  the  planet  casts  in  a  direction  opposite  to  the  sun.  The 
exact  moment  when  the  satellite  passes  into  the  shadow,  or  comes 
out  of  it,  is  calculated  by  astronomers.  But  sometimes  the  earth 
and  Jupiter  are  on  the  same  side,  and  sometimes  on  opposite  sides 
of  the  sun  ;  consequently,  the  earth  is,  in  the  former  case,  the 
whole  diameter  of  its  orbit,  or  about  one  hundred  and  eighty-five 
millions  of  miles  nearer  to  Jupiter  than  in  the  latter.  Now  it  is 
found  by  observation,  that  an  eclipse  of  one  of  the  satellites  is 
seen  about  sixteen  minutes  and  a  half  sooner  when  the  earth  is 
nearest  to  Jupiter,  than  when  it  is  most  remote  from  it,  and  con- 
sequently, the  light  must  occupy  this  time  in  passing  through  the 
diameter  of  the  earth's  orbit,  and  must  therefore  travel  at  the  rate 
of  about  186,868  miles  per  second,  according  to  this  determina- 
tion. 

Another  method  of  estimating  the  velocity  of  light,  wholly 
independent  of  the  preceding,  is  derived  from  what  is  called  the 
aberration  of  the  fixed  stars.  The  apparent  place  of  a  fixed  star 
is  altered  by  the  motion  of  its  light  being  combined  with  the  mo- 
tion of  the  earth  in  its  orbit.  The  place  of  a  luminous  object  is 
determined  by  the  direction  in  which  its  light  meets  the  eye.  But 
the  direction  of  the  impulse  of  light  on  the  eye  is  modified  by  the 
15 


•224  OPTICS. 

motion  of  the  observer  himself,  and  the  object  appears  forward  of 
its  true  place.  The  stars,  for  this  reason,  appear  slightly  displaced 
in  the  direction  in  which  the  earth  is  moving ;  and  the  velocity 
of  the  earth  being  known,  that  of  light  may  be  computed  in  the 
same  manner  as  we  determine  one  component,  when  the  angles 
and  the  other  component  are  known. 

342.  Determination  of  the  Velocity  of  Light  by  Ex- 
periment.— The  velocity  of  light  has  been  determined  also  by 
experiment,  in  a  manner  somewhat  analogous  to  that  employed 
by  Wheatstone  for  ascertaining  the  velocity  of  electricity.  The 
method  adopted  by  Foucault  is  essentially  the  following  : 

Through  an  aperture  A,  in  a  shutter  (Fig.  205),  a  beam  of 
light  is  admitted,  which  passing  through  an  inclined  transparent 

FIG.  205. 


glass  mirror,  E,  and  through  a  lens  of  very  long  focus,  B,  falls 
upon  a  mirror  C,  and  is  reflected  to  a  mirror  D ;  the  mirror  D 
again  reflects  the  beam  back  to  C,  whence  it  is  returned  through 
the  lens  B  to  the  glass  mirror  E,  is  reflected,  and  finally  enters 
the  eye.  The  mirror  #  is  so  mounted  as  to  rotate  with  great 
velocity  upon  an  axis,  perpendicular  to  the  plane  of  the  paper  in 
the  case  supposed. 

If  the  mirror  C  rotate  slowly,  in  the  direction  of  the  arrow, 
the  beam  will  alternately  disappear  and  reappear  at  the  point  a  ; 
but  if  the  velocity  be  increased  to  30  or  more  revolutions  per 
second  the  impression  on  the  eye  becomes  persistent  and  the 
beam  is  seen  without  interruptions,  appearing  stationary  at  a.  If 
now  the  speed  of  the  mirror  C  be  increased  to  from  300  to  600 
revolutions  per  second,  the  change  of  position  of  the  mirror  C, 
while  light  is  passing  from  it  to  D  and  back  again,  is  sufficient  to 
return  the  reflected  beam  to  some  point  b,  the  distance  from  a 
depending  upon  the  velocity  of  rotation.  From  the  displacement 
at  b,  the  angular  motion  of  the  mirror  at  C,  while  the  beam  tra- 
verses the  distance  from  CtoD  and  bacK  again,  can  be  determined ; 
and  knowing  the  rate  of  rotation,  this  fraction  of  one  turn  gives 


LOSS    OF    INTENSITY    BY    DISTANCE.  225 

the  time  which  the  light  required  to  traverse  double  the  distance 
C  D,  and  hence  its  velocity.  Such  is  an  outline  of  the  mode  of 
experimenting,  all  details  for  securing  precision  having  been 
omitted. 

By  this  method  the  velocity  given  in  Art.  341 — 186,300  miles 
per  second — was  determined  by  A.  A.  Michelson,  II.  S.  N.  The 
distance  between  the  revolving  mirror  C  and  the  mirror  D  was 
2000  feet. 

343.  Loss  of  Intensity  by  Distance. — The  intensity  of 
light  varies  inversely  as  the  square  of  the  distance.     In  Fig.  206, 
suppose  light  to  radi- 
ate from  S,  through 

the  rectangle  A  C, 
and  fall  on  EG,  paral- 
lel to  A  C.  AsSAE, 
8  B  F,  &c.,  are 
straight  lines,  the 
triangles,  8  A  B, 
8  E  F,  are  similar, 
as  also  the  rectangles,  A  C,  E  G ;  therefore,  A  C  :  E  G  : :  A  B*  : 
E  F*  : :  S  A2 :  S  Ez.  But  the  same  quantity  of  light,  being  diffused 
over  A  C  and  E  G,  will  be  more  intense,  as  the  surface  is  smaller. 
Hence,  the  intensity  of  light  at  E  :  intensity  at  A  : :  A  C  :  E  G  : : 
JS  A*  :  S  E*,  which  proves  the  proposition.  This  demonstration 
is  applicable  to  every  kind  of  emanation  in  straight  lines  from  a 
point. 

If  the  surface  receiving  the  light  be  oblique  to  the  axis  of  the 
beam,  the  intensity  of  illumination  is  proportional  to  the  sine  of 
the  angle  which  the  rays  make  with  the  surface.  Let  Fig.  207 
represent  a  section  through  the  axis  ^  3Q7 

of  a  beam  passing  through  an  orifice 
a  ft  and  falling  upon  the  inclined 

surface  at  c  d.     Now  because  of  the     _^ of 

obliquity,  the  surface  c  d  is  greater    •  ^> 

than  the  section  at  right  angles  to 

the  beam  represented  by  e  c,  and 

hence  is  less  intensely  illuminated 

at  any  point.     But  surface  e  c  :  surface  c  d  : :  line  e  c  :  line  c  d  : : 

sine  e  d  c  :  sine  ced. 

Calling  the  illumination  upon  any  point  of  the  right  section 
<e  c  unity  or  u,  the  illumination  upon  any  point  of  c  d  will  be 

e  c 

u  x  — 7  =  u  x  sine  e  dc. 
cd 

344.  Brightness  the  Same  at  all  Distances.— The  bright- 


226  OPTICS. 

ness  of  an  object  is  the  quantity  of  light  which  it  sheds,  as  com- 
pared with  the  apparent  area  from  which  it  comes.  Now  the  quan- 
tity (or  intensity),  as  has  just  been  shown,  varies  inversely  as  the 
square  of  the  distance.  The  apparent  area  of  a  given  surface  also 
diminishes  in  the  same  ratio,  as  we  recede  from  it.  Hence  the 
brightness  is  constant.  For  illustration,  if  we  remove  to  three 
times  the  distance  from  a  luminous  body,  we  receive  into  the  eye 
nine  times  less  light,  but  the  body  also  appears  nine  times  smaller, 
so  that  the  relation  of  light  to  apparent  area  remains  the  same. 

345.  Bunsen's  Photometer. — Photometers  are  instruments 
for  determining  the  relative  intensities  of  two  sources  of  light. 
The  Bunsen  (Fig.  208)  photometer  makes  use  of  the  fact,  that  a 
grease  spot,  upon  a  piece  of  bibulous  white  paper,  appears  darker 
than  the  surrounding  paper,  if  it  be  more  intensely  illuminated  on 
the  side  toward  the  observer ;  and  appears  lighter  than  the  paper, 
if  more  intensely  illuminated  on  the  side  away  from  the  observer. 
When  equally  illuminated  on  both  sides,  the  spot  is  invisible. 
The  two  lights  to  be  compared  are  placed  on  opposite  sides  of  the 
FIG.  208. 


grease-spot  screen.  One  of  them  is  then  moved  until  the  spot  be- 
comes invisible,  then  the  intensities  of  the  two  lights  are  as  the 
squares  of  their  distances  from  the  spot.  For,  suppose  that  one 
of  the  lights  is  a  standard  candle,  and  that  it  is  placed  at  a  unit's 
distance  from  the  screen.  Any  other  light,  at  the  same  distance, 
illuminating  the  screen  with  the  same  intensity,  would  have  an  in- 
tensity of  one  candle  power.  If,  at  the  same  distance,  the  illumi- 
nation were  16  times  as  intense,  then  this  light  has  an  intensity  of 
16  candle  power.  Now,  the  eye  cannot  estimate  the  value  of  dif- 
ferent intensities.  By  the  grease  spot,  however,  it  can  tell  when 
two  intensities  ai*e  equal.  Suppose  that  a  candle  at  unit's  dis- 
tance is  balanced  by  an  incandescent  lamp  at  4  units'  distance. 
From  Art.  343  we  know  that  the  lamp  would  illuminate  the  screen 
with  16  times  the  intensity  if  it  were  moved  up  to  a  unit's  distance. 
It  is,  then,  a  16  (=  4s)  candle-power  lamp. 

The  distance  of  the  standard  light  from  the  screen  may  always 


SHADOWS.  227 

be  considered  as  unity.     Then  the  intensity  of  the  other  is  equal 
to  the  square  of  its  distance  (measured  in  this  unit)  from  the  screen. 

346.  Rumford's  Photometer. — Let  the  two  unequal  lights 
be  placed  at  A  and  B  (Fig.  209)  so  that  the  shadows  of  an  opaque 
rod  C  shall  fall  side  by  side  upon  a  screen,  as  at  a  and  b.  The 
portion  of  the  screen  upon  which  the  shadow  a  falls  receives  light 
only  from  the  candle  B  and  none  from  the  gas  flame  A  ;  the  por- 
tion b  is  illuminated  by  A  alone.  The  opaque  body  thus  secures 
for  each  light  a  portion  of  the  screen  which  it  alone  illuminates. 

FIG.  209. 


Now  move  either  light  towards  or  from  the  screen  until  the  two 
portions  a  and  b  are  equally  illuminated  by  their  respective  lights, 
and  then  measure  the  distances  from  A  to  b,  =  m,  and  from  B  to 
a,  =  n.  B  at  distance  n  illuminates  the  screen  as  intensely  as  A 
at  distance  m. 

Then,  as  in  Art.  345,  A  :  B  =  m*  :  n* ; 

or,  the  intensities  vary  directly  as  the  squares  of  those  distances  from 
the  screens  at  which  equal  illumination  is  obtained. 

347.  Shadows. — When  a  luminous  body  shines  on  one  which 
is  opaque,  the  space  beyond  the  latter,  from  which  the  light  is 
excluded,  is  called  a  shadow.  The  same  word,  as  commonly  used, 
denotes  only  the  section  of  a  shadow  made  by  a  surface  which 
crosses  it.  Shadows  are  either  total  or  partial.  If  tangents  are 
drawn  on  all  the  corresponding  sides  of  the  two  bodies,  the  space 
inclosed  by  them  beyond  the  opaque  body  is  the  total  shadow  ;  if 
other  tangents  are  drawn,  crossing  each  other  between  the  bodies, 
the  space  between  the  total  shadow  and  the  latter  system  of  tan- 
gents is  the  partial  shadow,  or  penumbra.  In  case  the  bodies  are 
spheres,  as  in  Fig.  210,  the  total  shadow  will  be  a  cylinder,  or  con- 
ical frustum,  each  of  infinite  length,  or  a  complete  cone,  accord- 
ing to  the  relative  size  of  the  spheres.  But,  in  every  case,  the 
penumbra  and  inclosed  total  shadow  will  form  an  increasing 
frustum.  It  is  obvious  that  the  shade  of  the  penumbra  grows. 


228 


OPTICS. 


gradually  deeper  from  the  outer  surface  to   the  total   shadow 
within  it. 

Every  shadow  cast  by  the  sun  has  a  penumbra  bordering  it, 
which  gives  to  the  shadow  an  ill-defined  edge ;  and  the  more 


Flo.  210. 


remote  the  sectional  shadow  is  from  the  opaque  body  which  casts 
it,  the  broader 'will  be  the  partial  shadow  on  the  edge. 

If  instead  of  a  luminous  body  of  sensible  magnitude,  the 
source  of  light  be  a  point,  then  no  penumbra  will  be  formed. 
The  electric  arc  between  carbon  points  casts  a  sharply-defined 
shadow  of  a  hair  upon  a  screen  placed  at  a  great  distance. 


CHAPTER    II. 

REFLECTION  OF  LIGHT. 

348.  Radiant  and  Specular  Reflection.— Light  is  said  to 
be  reflected  when,  on  meeting  a  surface,  it  is  turned  back  into  the 
same  medium.  In  ordinary  cases  of  reflection,  the  light  is  diffused 
in  all  directions,  and  it  is  by  means  of  the  light  thus  scattered 
from  a  body  that  it  becomes  visible,  when  it  sheds  no  light  of  its 
own.  This  is  called  radiant  reflection.  It  is  produced  by  unpol- 
ished surfaces.  But  when  a  surface  is  highly  polished,  a  beam  of 
light  falling  on  it  is  reflected  in  some  particular  direction  ;  and, 
if  the  eye  is  placed  in  this  reflected  beam,  it  is  not  the  reflecting 
surface  which  is  seen,  but  the  original  object,  apparently  in  a  new 
position.  This  is  called  specular  reflection.  It  is,  however,  gener- 
ally accompanied  by  some  degree  of  radiant  reflection,  since  the 
reflector  itself  is  commonly  visible  in  all  directions.  Ordinary 


LAW     OF    REFLECTION. 


229 


mirrors  are  not  suitable  for  accurate  experiments  on  reflection, 
because  light  is  modified  by  the  glass  through  which  it  passes. 
The  speculum  is  therefore  used,  which  is  a  reflector  made  of  solid 
metal,  and  accurately  ground  to  any  required  form,  either  plane, 
convex,  or  concave.  The  word  mirror  is,  however,  much  used  in 
optics  for  every  kind  of  reflector. 

Optical  experiments  are  usually  performed  on  a  beam  of  light 
admitted  through  an  aperture  into  a  darkened  room;  the  direc- 
tion of  the  beam  being  regulated  by  an  adjustable  mirror  placed 
outside.  An  instrument  consisting  of  a  plane  speculum  moved  by 
a  clock,  in  such  a  manner  that  the  reflected  sunbeam  shall  remain 
stationary  at  all  hours  of  the  day,  is  called  a  heliostat. 

349.  The  Law  of  Reflection. — When  a  ray  of  light  is  inci- 
dent on  a  mirror,  the  angle  between  it  and  a  perpendicular  to  the 
surface  at  the  point  of  incidence,  is  called  the  angle  of  incidence; 
and  the  angle  between  the  reflected  ray  and  the  same  perpendicu- 
lar, is  called  the  angle  of  reflection.  The  law  of  reflection  found 
to  be  universally  true  is  the  following : 

The  incident  ray,  the  reflected  ray,  and  the  normal  to  the  sur- 
face are  in  the  same  plane,  and  the  normal  bisects  the  angle  which 
these  rays  make  with  each  other. 

This  is  well  shown  by  attaching  a  small  mirror  to  the  centre 
of  a  graduated  semicircle  perpendicular  to  its  plane.  Let  M  D  N 
(Fig.  211)  be  the  semicircle,  graduated  from  D  both  ways  to  M 
and  N,  and  mounted  so  that  it 
can  be  revolved  on  its  centre,  and 
clamped  in  any  position.  Let  the 
small  mirror  be  at  C,  with  its  plane 
perpendicular  to  C  D  ;  then  a  ray 
from  the  heliostat,  as  A  C,  passing 
the  edge  at  a  particular  degree, 
will  be  seen  after  reflection  to  pass 
the  corresponding  degree  in  the 
other  quadrant.  By  revolving  the 
semicircle,  any  angle  of  incidence 
may  be  tried,  and  the  two  rays  are 
always  found  to  be  in  the  same 
plane  with  0  D,  and  equally  in- 
clined to  it. 

As  the  mirror  revolves,  the  re- 
flected ray  revolves  twice  as  fast. 

For  A  C  D  is  increased  or  diminished  by  the  angle  through 
which  the  mirror  turns ;  therefore  D  C  B  is  also  increased  or 
diminished  by  the  same  j  hence  A  C  B,  the  angle  between  the  two 


FIG.  211. 


230  OPTICS. 

rays,  is  increased  or  diminished  by  the  sum  of  both,  or  twice  the 
same  angle. 

It  follows  from  the  law  of  reflection,  that  a  ray  which  falls  on 
a  mirror  perpendicularly,  retraces  its  own  path  after  reflection.  It 
is  obvious,  also,  that  the  complements  of  the  angles  of  incidence 
and  reflection  are  equal,  i.  e.  A  C  M  =  B  C  N.  The  law  of  reflec- 
tion is  applicable  to  curved  as  well  as  to  plane  mirrors  ;  the  radius 
of  curvature  at  any  point  being  the  perpendicular  with  which  the 
incident  and  reflected  rays  make  equal  angles. 

Radiant  reflection  forms  no  exception  to  the  foregoing  law, 
though  the  incident  rays  are  in  one  and  the  same  direction,  and 
the  reflected  rays  are  scattered  every  way.  For  the  minute  cavi- 
ties and  prominences  which  constitute  the  roughness  of  the  gen- 
eral surface  are  bounded  by  small  surfaces  lying  at  all  inclinations ; 
and  each  one  reflecting  the  rays  which  meet  it  in  accordance  with 
the  law,  those  rays  are  necessarily  thrown  off  in  all  possible  direc- 
tions. 

The  proportion  of  the  incident  light  reflected  varies  with 
the  angle  of  incidence.  When  light  strikes  the  surface  of  water 
perpendicularly  only  .018  is  reflected,  the  rest  entering  the  water  ; 
but  at  an  incidence  of  89£°  .  721  is  reflected.  In  the  case  of  mer- 
cury .666  is  reflected  at  perpendicular  incidence,  while  .721  is 
reflected  at  89£°,  the  non-reflected  rays  entering  the  metal  and 
being  destroyed,  as  light. 

350.  Inclination  of  Rays  to  each  other  not  altered  by 
the  Plane  Mirror. — 

1.  Rays  which  diverge  before  reflection,  diverge  at  the  same 
angle  after  reflection. 

Let  M  N  (Fig.  212)  be  a  plane  mirror,  and  A  B,  A  C,  any  two 
rays  of  light  falling  upon  it  from  the 
radiant  A,  and  reflected  in  the  lines 
BE,  C  G.  Draw  the  perpendicular 
A  P,  and  produce  it  indefinitely,  as  to  F, 
behind  the  mirror;  also  produce  the  re- 
flected rays  back  of  the  mirror.  Let  Q  R 
be  perpendicular  to  the  mirror  at  the 
point  B ;  it  is  therefore  parallel  to  A  F, 
and  the  plane  passing  through  A  F  and 
Q  R,  is  that  which  includes  the  ray 
A  B,  B  E.  Therefore  E  B,  when  pro- 
duced back  of  the  mirror,  intersects  A  P 

produced.  Let  .Fbe  the  point  of  intersection.  B  A  F  =  A  B  Q, 
and  A  FB  =  EB  Q;  but  A  B  Q  =  E  B  Q  (Art.  349)  ;  .-.  BA  F 
=  A  FBS  and  A  B  —  F  B.  If  P  and  B  be  joined,  P  B  being  in 


SPHERICAL    MIRRORS.  231 

the  plane  M  N  is  perpendicular  to  A  F,  and  therefore  bisects  it. 
Hence,  the  reflected  ray  meets  the  perpendicular  A  Fas,  far  behind 
the  mirror,  as  the  incident  ray  does  in  front.  In  the  same  way  it 
may  be  proved  that  A  C  =  C  F,  and  that  C  G,  when  produced 
back  of  the  mirror,  meets  A  F  at  the  same  point  F. 

Now,  since  the  triangles  A  C B  and  FOB,  have  their  sides 
respectively  equal,  their  angles  are  equal  also ;  hence  B  A  C  = 
B  F  C.  Therefore  any  two  rays  diverge  at  the  same  angle  after 
reflection  as  they  did  before  reflection. 

Since  the  reflected  rays  seem  to  emanate  from  F,  that  point  is 
called  the  apparent  radiant ;  A  is  the  real  radiant. 

2.  Eays  which  converge  before  reflection,  converge  at  the  same 
.angle  after  reflection.     Let  E  B,  G  C,  be  incident  ray?  converging 
toward  F,  and  let  B  A,  C  A,  be  the  reflected  rays.     It  may  be 
proved  as  before,  that  A  and  Fare  in  the  same  perpendicular, 
A  F,  and  equidistant  from  P,  and  that  E  F  G  =  B  A  C. 

The  point  F,  to  which  the  incident  rays  were  converging,  is 
-called  the  virtual  focus  ;  A  is  the  real  focus. 

3.  Rays  which  are  parallel  before  reflection  are  parallel  after 
reflection. 

It  has  been  proved  in  case  1,  that  F,  the  intersection  of  the 
reflected  rays,  is  as  far  behind  the  mirror,  as  A,  the  intersection 
of  incident  rays,  is  before  it.  Now,  if  the  incident  rays  are  parallel, 
A  is  at  an  infinite  distance  from  the  mirror.  Therefore  F  is  at 
an  infinite  distance  behind  it,  and  the  reflected  rays  are  parallel. 

In  all  cases,  therefore,  rays  reflected  by  a  plane  mirror  retain 
the  same  inclination  to  each  other  which  they  had  before  reflec- 
tion. 

351.  Spherical  Mirrors. — A  spherical  mirror  is  one  which 
forms  a  part  of  the  surface  of  a  sphere,  and  is  either  convex  or 
concave.     The  axis  of  such  a  mirror  is  that  radius  of  the  sphere 
which  passes  through  the  middle  of  the  mirror.     In  the  practical 
use  of  spherical  mirrors,  it  is  found  that  the  light  must  strike  the 
surface  very  nearly  at  right  angles ;  hence,  in  the  following  state- 
ments, the  mirror  is  supposed  to  be  a  very  small  part  of  the  whole 
spherical  surface,  and  the  rays  nearly  coincident  with  the  axis. 

It  is  sufficient  to  trace  the  course  of  the  rays  on  one  side  of  the 
axis,  since,  on  account  of  the  symmetry  of  the  mirror  around  the 
axis,  the  same  effect  is  produced  on  every  side. 

352.  Converging  Effect  of  a  Concave  Mirror. — 

1.  Parallel  rays  are  converged  to  the  middle  point  between  the 
centre  and  surface,  which  is  therefore  called  the  focus  of  parallel 
rays  or  the  principal  focus.  Let  R  A,  L  E  (Fig.  213),  be  parallel 


232 


OPTICS. 


rays  incident  upon  the  concave  mirror  A  B,  whose  centre  of  con- 
cavity is  C.     The  ray  L  E,  passing  through   C,  and  therefore. 


FIG.  213. 


perpendicular  to  the  mirror  at  E,  is  reflected  directly  back.  Join 
C  A,  and  make  C  A  F  =  R  A  C-,  then  R  A  is  reflected  in  the 
line  A  F,  and  the  two  reflected  rays  meet  at  F.  R  A  C  =  AC  F, 
.:  A  C  F  =  FA  C,  and.  A  F  =  ~C  F,  and  as  A  and  E  are  very 
near  together,  E  F '=  A  F  =  F  (7;  that  is,  the  focus  qf  parallel 
rays  is  at  the  middle  point  between  C  and  E. 

2.  Diverging  rays,  falling  on  a  given  concave  mirror,  are  re- 
flected converging,  parallel,  or  less  diverging,  according  to  the 
degree  of  divergency  in  the  original  pencil.  Let  C  (Fig.  214)  be 
the  centre  of  concavity,  and  F  the  focus  of  parallel  rays.  Then, 

FIG.  214. 


rays  diverging  from  any  point,  A,  beyond  C,  will  be  converged  to- 
some  point,  a,  between  G  and  F,  since  the  angles  of  incidence  and 
reflection  are  less  than  those  for  parallel  rays.  Rays  diverging 
from  C  are  reflected  back  to  C ;  those  from  points  between  C  and 
F,  as  a,  are  converged  to  points  beyond  C,  as  A  ;  those  diverging 
from  F  become  parallel ;  and  those  from  points  between  F  and  the 
mirror,  as  D,  diverge  after  reflection,  but  at  a  less  angle  than  be- 
fore, and  seem  to  flow  from  A'.  To  prove,  in  the  last  case,  that 
the  angle  of  divergence,  A',  after  reflection,  is  less  than  the  angle 
D,  the  divergence  before  reflection,  observe  that  the  angle  A'  is 
less  than  the  exterior  angle  HBO,  or  its  equal,  DBG  (Art. 
349)  ;  and  D  B  (7 is  less  than  the  exterior,  A'  D  B;  much  more, 
then,  is  A1  less  than  A'  D  B. 

3.  Converging  rays  are  made  to  converge  more.  The  rays  H  B>. 
A  E,  converging  to  A',  are  reflected  to  D,  nearer  the  mirror  than 
Fis.  And  it  has  been  shown  that  the  angle  D  is  larger  than  A', 
hence  the  convergency  is  increased. 

From  the  three  foregoing  cases,  it  appears  that  the  concave 


CONJUGATE    FOCI.  233 

mirror  always  tends  to  produce  convergency ;  since,  when  it  does 
not  actually  produce  it,  it  diminishes  divergency. 

The  principal  focus  can  be  determined  practically  by  receiving 
the  sun's  rays  upon  the  mirror,  parallel  to  its  axis,  and  finding 
the  point  at  which  a  sharp  image  of  the  sun  is  formed.  The  dis- 
tance of  this  image  from  the  surface  is  one-half  the  radius  of  cur- 
vature. 

353.  Conjugate  Foci. — When  light  radiates  from  A,  it  is 
reflected  to  a  (Fig.  214) ;  when  from  a,  it  meets  at  A.     Any  two 
such  interchangeable  points  are  called  conjugate  foci.    If  the  radius 
of  the  mirror  and  the  distance  of  one  focus  from  the  mirror  are 
given,   the  distance  of  its  conjugate  focus  may  be  determined. 
Let  the  radius  =  r ;  the  distance  A  E  =  m  :  and  a  E  =  n.     As 
the  angle  A  B  a  is  bisected  by  B  C,  A  B  :  a  B  : :  A  0  :  a  C ;  that 
is,  since  £  E  is  very  small,  A  E :  a  E : :  A  C :  a  C,  or,  m  :  n  : : 
m  —  r  :  r  —  n. 

n  r  m  r 

.'.  m  = ;  and  n  =  -? . 

2  n  —  r  2m  —  r 

If  A  is  not  on  the  axis  of  the  mirror,  as  in  Fig.  215,  let  a  line 
be  drawn  through  A  and  C,  meet- 
ing the  mirror  in  E ;  this  is  called 
a  secondary  axis,  and  the  light 
radiating  from  A  will  be  reflected 
to  a  on  the  same  secondary  axis, 
for  A  E  is  perpendicular  to  the 

mirror,  and  will  be  reflected  directly  back  ;  and  if  A  E  and  0  E 
are  given,  a  E  may  be  found  as  before. 

354.  Diverging  Effect  of  a  Convex  Mirror.— 

1.  Parallel  rays  are  reflected  diverging  from  the  middle  point 
between  the  centre  and  surface.  Let  C  (Fig.  216)  be  the  centre 
of  convexity  of  the  mirror  M  N,  and  draw  the  radii,  CM,  CD,. 

FIG.  216. 


producing  them  in  front  of  the  mirror  ;  these  are  perpendicular  to 
the  surface.  The  ray  R  D  will  be  reflected  back  ;  A  M  will  be 
reflected  in  M  B,  making  B  M  E  =  A  M  E.  Produce  the  re- 
flected ray  back  of  the  mirror,  and  it  will  meet  the  axis  in  F,  mid- 
way from  CtoD;  for  FCM  =  A  ME,  and  FM  C=  B  M  E-, 
therefore  the  triangle  F  C  M  is  isosceles,  and  C  F—  F  M,  and  as 


234: 


OPTICS. 


3/is  very  near  D,  C  F  =  F  D.  Hence  the  rays,  after  reflection, 
diverge  as  if  they  radiated  from  a  point  in  the  middle  of  C  D, 
which  is  the  apparent  radiant. 

2.  Diverging  rays  have  their  divergency  increased.     Let  A  D, 
A  M  (Fig.  217),  be  the  diverging  rays ;  D  A,  M  B,  the  reflected 
FIG.  217. 


rays ;  these  when  produced  meet  at  F,  which  is  the  apparent  radi- 
ant. MAFis  the  divergency  of  the  incident  rays,  and  A  F  B 
of  the  reflected  rays.  Now  the  exterior  angle,  A  F  B,  is  greater 
than  C  M  F,  or  B  M  E,  or  A  ME.  But  A  M  E,  being  exterior,  is 
greater  than  M  A  F;  much  more,  then,  is  A  FB  greater  than  MA  F. 
3.  Convergent  rays  are  at  least  rendered  less  convergent,  and 
may  become  parallel  or  divergent,  according  to  the  degree  of  pre- 
vious convergency.  The  two  first  effects  are  shown  by  Figs.  216 
and  217,  reversing  the  order  of  the  rays.  And  it  is  easy  to  per- 
ceive that  rays  converging  to  C,  will  diverge  from  C  after  reflec- 
tion ;  if  to  a  point  more  distant  than  C,  they  will  diverge  after- 
ward from  a  point  between  C  and  F  (Fig.  216),  and  vice  versa. 

The  general  effect,  therefore,  of  a  convex  mirror,  is  to  produce 
divergency. 

A  and  F  (Fig.  217)  are  called  conjugate  foci,  being  inter- 
changeable points  ;  for  rays  from  A  move  after  reflection  as  though 
from  F,  and  rays  converging  to  F  are  by  reflection  converged  to 
A.  Conjugate  foci,  in  the  case  of  the  convex  mirror,  are  in  the 
same  axis  either  principal  or  secondary,  as  they  are  in  the  concave 

mirror,  and  for  the  same  reason, 
viz.,  that  every  axis  is  perpendic- 
ular to  the  surface. 

Their  relative  positions  may 
be  determined  by  the  formula, 
-^ ^=^  easily  deduced,  as  in  Art.  353, 

.(7   m=      nr 

To  determine  the  radius  of  cur- 
vature experimentally:  Through 
a  circular  opening  in  a  screen 
whose  diameter  is  greater  than 
E  H  (Fig.- 218),  receive  the  sun's 
rays  upon  the  mirror,  parallel  to  the  axis,  and  move  the  screen  so 


FIG.  218. 


r 


IMAGES     BY    A     PLANE    MIRROR.  235 

that  the  diameter  KD  of  the  illuminated  circle  is  twice  the  chord 
E  H  of  the  mirror  ;  then  measure 


To  render  this  method  more  accurate  cover  all  of  the  mirror, 
except  a  small  central  circle,  with  some  opaque  covering,  and  use 
only  the  exposed  portion  as  above.  I 

355.  Images  by  Reflection.  —  An  optical  image  consists  of 
a  collection  of  focal  points,  from  which  light  either  really  or  appa- 
rently radiates.     When  rays  are  converged  to  a  focus  they  do  not 
stop,  but  cross,  and  diverge  again,  as  if  originally  emanating  from 
the  focal  point.     A  collection  of  such  points,  arranged  in  order, 
constitutes  a  real  image.     When  rays  are  reflected  diverging,  they 
proceed  as  though  they  emanated  from  a  point  behind  the  mirror. 
A  collection  of  such  imaginary  radiants  forms  an  apparent  or 
virtual  image.   The  images  formed  by  plane  and  convex  mirrors  are 
always  apparent  ;  those  formed  by  concave  mirrors  may  be  of  either 
kind. 

356.  Images  by  a  Plane  Mirror.  —  When  an  object  is  before 
a  plane  mirror,  its  image  is  at  the  same  distance  behind  it,  of  the 
same  magnitude,  and  equally  inclined  to  it.     Let  M  JV  (Fig.  219) 
be  a  plane  mirror,  and  A  B  an  ob- 

ject before  it,  and  Jet  the  position  FIG.  219. 

of  the  object  be  such  that  the  re- 

flected   rays  may    enter    the    eye 

placed  at  H.     From  A  and  B  let 

fall  upon  the  plane  of  the  mirror 

the  perpendiculars  A  E,  B  G,  and 

produce  them,  making  E  a  =  AE, 

and  G  b  —  B  G.     Now,  since  the 

rays  from  A  will,  after  reflection, 

radiate  as  if  from  a  (Art.  350),  and  those  from  B,  as  if  from  J,  and 

the  same  of  all  other  points,  therefore  the  image  and  object  are 

equally  distant  from  the  mirror.     A  C,  a  c,  parallel  to  the  mirror, 

are  equal  ;    as  B  G  =  b  G,  and  A  E  —  a  E,  therefore,  by  sub- 

traction, B  C  =  b  c  ;  also  the  right  angles  at  C  and  c  are  equal. 

Therefore  A  B  —  a  b,  and  B  A  G  —  b  a  c;  that  is,  the  object  and 

image  are  of  equal  size,,  and  equally  inclined  to  the  mirror. 

It  appears  from  the  demonstration,  that  the  object  and  its 
image  are  comprehended  between  the  same  perpendiculars  to  the 
plane  of  the  mirror;  and  this  image  will  appear  in  the  same 
position  whatever  may  be  the  position  of  the  eye. 

The  object  and  image  obviously  have  to  each  other  twice  the 
inclination  that  each  has  to  the  mirror.  Hence,  in  a  mirror  in- 
clined 45°  to  the  horizon,  a  horizontal  surface  appears  vertical, 
and  one  which  is  vertical  appears  horizontal. 


236 


OPTICS. 


FIG.  220. 


357.  Symmetry  of  Object  and  Image. — All  the  three  di- 
mensions of  the  object  and  image  are  respectively  equal,  as  shown 
above,  but  one  of  them  is  inverted  in  position,  namely,  that  dimen- 
sion which  is  perpendicular  to  the  mirror.     Hence,  a  person  and 
his  image  face  in  opposite  directions  ;  and  trees  seen  in  a  lake  have 
their  tops  downward.     Those  dimensions  which  are  parallel  to  the 
mirror  are  not  inverted.     In  consequence  of  the  inversion  of  one 
dimension  alone,  the  object  and  its  image  are  not  similar,  but 
symmetrical  forms ;  and  one  could  not  coincide  with  the  other  if 
brought  to  occupy  the  same  space.    The  image  of  a  right  hand  is 
a  left  hand,  and  all  relations  of  right  and  left  are  reversed.     It  is 
for  this  reason  that  a  printed  page,  seen  in  a  mirror,  is  like  the 
type  with  which  it  was  printed. 

358.  The  Length  of  Mirror  Requisite  for  Seeing  an 
Object. — If  an  object  is  parallel  to  a  mirror,  the  length  of  mirror 
occupied  by  the  image  is  to  the  length  of  the  object  as  the  reflected 

ray  to  the  sum  of  the  incident  and  reflected 
rays.  Let  A  B  (Fig.  220)  be  the  length  of  the 
object,  CD  that  of  the  image,  and  F  G  that  of 
the  space  occupied  on  the  mirror;  then,  by 
similar  triangles,  F  G  :  CD  : :  EF :  E  C.  But 
CD  =  A  B,  and  CF=  A  F;  .-.  FG  :  A  B:: 
E  F  :A  F  +  F  E.  If  the  eye  is  brought 
nearer  the  mirror,  the  space  on  the  mirror 
occupied  by  the  image  is  diminished,  because 
to  A  F+  F  E  a  less  ratio  than  before.  The  same  effect  is 

produced  by  removing  the  object 

further  from   the  mirror.     The 

length  of  mirror  necessary  for  a 

person  to  see  himself  is  equal  to 

half  his  height,  because  in  that 

case,  E  F:A  F+  FE::l  :  2, 

which  ratio  will  not  be  altered 

by  change  of  distance. 

359.  Displacement  of  Im- 
age by  Two  Reflections. — If 

an  image  is  seen  by  light  reflected 
from  two  mirrors  in  a  plane  per- 
pendicular to  their  common  sec- 
tion, its  angular  deviation  from 
the  object  is  equal  to  twice  the 
inclination  of  the  mirrors.  Let 
AB,CD  (Fig.  221)  be  two  plane 


FIG.  221. 


MULTIPLIED    IMAGES    BY    TWO    MIRRORS.      237 

mirrors  inclined  at  the  angle  A  G  C.  If  an  eye  at  H  sees  the  star 
S  in  the  direction  0,  the  angle  8  H  0  =  2  A  G  C. 

For  the  exterior  angle  C  D  B  =  b  —  a  +  G,  or  2  b  =  2  a  + 
2G,an&BDO=2b  =  2a+H;  hence  2a  +  H=2a  +  2G; 
therefore  H  —  2  G. 

This  principle  is  employed  in  the  construction  of  Hadley's 
quadrant,  and  the  sextant,  used  at  sea  for  measuring  angular  dis- 
tances. The  angles  measured  are  twice  as  great  as  the  arc  passed 
over  by  the  index  which  carries  the  revolving  mirror ;  hence,  in 
the  quadrant,  an  arc  of  45°  is  graduated  into  90° ;  and,  in  the  sex- 
tant, an  arc  of  60°  is  graduated  into  120°. 

360.  Multiplied  Images  by  Two  Mirrors. — 

1.  Parallel  Mirrors.  The  series  of  images  is  infinite  in  num- 
ber, and  arranged  in  a  straight  line,  perpendicular  to  the  mirrors. 
The  object  a,  between  the  parallel  mirrors,  A  and  B  (Fig.  222), 
has  an  image  at  a',  as  far  behind  A  as  a  is  in  front  of  it.  To 

FIG.  222. 


I" 


avoid  confusion,  a  pencil  from  only  one  point  o  is  drawn,  once  re- 
flected at  c,  and  entering  the  eye  as  though  it  came  from  o'.  The 
rays  reflected  by  A  diverge  as  though  they  emanated  from  d\ 
hence,  the  light  reflected  from  A  upon  B  may  be  regarded  as  pro- 
ceeding from  a  real  object  at  a',  whose  image  will  be  b,  as  far  back 
of  B  as  a'  is  in  front  of  B.  The  light  reflected  from  B  to  A  again 
diverges  as  though  it  really  came  from  b,  and  regarding  b  as  a 
real  object  as  before  its  image  would  be  formed  at  a"  as  far  behind 
A  as  b  is  in  front  of  it.  The  pencil  which  enters  the  eye  seems 
to  proceed  from  o'",  having  been  reflected  from  e",  as  though  it 
came  from  o",  its  reflection  in  this  case  having  been  from  e'  a» 
though  it  came  from  o',  though  it  was  really  reflected  from  e  after 
having  emanated  from  o.  The  pencil  which  would  enter  the  eye 
from  a  third  image  at  the  left  of  a"  may  be  traced  through  all  its 
reflections  in  like  manner.  As  light  is  absorbed  and  scattered  at 
each  reflection  the  number  of  such  images  is  limited. 

The  multiplied  images  of  a  small  bright  object,  sometimes 


238 


OPTICS. 


seen  in  a  looking-glass,  are  produced  by  repeated  reflections  be- 
tween the  front  and  the  silvered  covering  on  the  back  side.  At 
each  internal  impact  cm  the  first  surface  some  light  escapes,  and 
shows  us  an  image,  while  another  portion  is  reflected  to  the  back, 
and  thence  forward  again.  The  image  of  a  lamp  viewed  very 
obliquely  in  a  mirror  is  sometimes  repeated  eight  or  ten  times  ;  and 
a  planet,  or  bright  star,  when  seen  in  a  looking-glass,  will  be  ac- 
companied by  three  or  four  faint  images,  caused  in  the  same  way. 
2.  Inclined  Mirrors.  Let  Q  (Fig.  223)  be  the  object,  and  0 
the  position  of  the  eye.  With  R  as  a  centre  and  radius  R  Q,  de- 
scribe a  circumference. 

Suppose  a  chord  Q  A  to  be  drawn  perpendicular  to  the  mirror 

8,    then   A   will    be    the 

PIG.  223.  image  of    Q.     Regarding 

A  as  a  real  object,  as  in 
the  case  of  parallel  mirrors, 
draw  a  chord  A  B  per- 
pendicular to  the  mirror 
T,  then  since  o  B  =  o  A, 
B  will  be  the  image  of  A. 
Suppose  a  chord  B  Cto  be 
drawn  perpendicular  to  the 
mirror  S,  then  C,  being 
as  far  behind  the  mir- 
ror 8  as  the  object  B, 
assumed  as  real,  is  in  front 
of  it,  will  be  the  image  of 
B ;  and  for  like  reasons 
D  will  be  the  image  of  C 
in  mirror  T.  All  the 
images  formed  by  the  inclined  mirrors  are  thus  seen  to  be  confined 
to  the  circumference  of  a  circle  described  as  above  stated.  There 
can  be  no  image  of  D,  since  it  lies  behind  both  mirrors  prolonged. 
The  image  A  is  seen  by  rays  which  proceed  from  Q  to  a  and  thence 
to  the  eye  at  0.  B  is  seen  as  though  the  rays  came  from  B  to  0, 
these  having  been  reflected  at  c  as  though  they  came  from  A,  the 
reflection  at  b  being  direct  from  Q.  The  image  C  is  seen  by  rays 
reflected  from  the  points  /,  e,  d;  and  D  by  rays  reflected  from 
k,  i,  h,  g.  The  reflections  occur  in  the  order  d,  e,f,  and  g,  h,  i,  k. 
Only  images  formed  by  light  first  reflected  from  8  have  been 
considered ;  a  second  series  produced  by  light  first  reflected  from 
T  may  .be  constructed  in  like  manner. 

361.  The  Kaleidoscope. — This  instrument,  when  carefully 
constructed,  beautifully  exhibits  the  phenomenon  of  multiplied 


IMAGES    BY    THE    CONCAVE    MIRROR.          239 

reflection  by  inclined  mirrors.  It  consists  of  a  tube  containing 
two  long,  narrow;  metallic  mirrors,  inclined  at  a  suitable  angle ; 
and  is  used  by  placing  the  objects  (fragments  of  colored  glass, 
&c.)  at  one  end,  and  applying  the  eye  to  the  other.  In  order  that 
there  may  be  perfect  symmetry  in  the  figure  made  up  of  the  ob- 
jects and  their  successive  images,  the  angle  of  the  mirrors  should 
be  of  such  size,  that  it  can  be  exactly  contained  an  even  number 
of  times  in  360°.  The  best  inclination  is  30° ;  and  the  field  of 
view  is  then  composed  of  12  sectors.  It  is  also  essential,  that  the 
small  objects  forming  the  picture,  should  lie  at  the  least  possible 
distance  beyond  the  mirrors.  To  insert  three  mirrors  instead  of 
two,  as  is  often  done,  only  serves  to  confuse  the  picture,  and  mar 
its  beauty. 

362.  Images  by  the  Concave  Mirror. — The  concave 
mirror  forms  various  images,  either  real  or  apparent,  either 
greater  or  less  than  the  object,  either  erect  or  inverted,  according 
to  the  place  of  the  object. 

1.  The  object  between  the  mirror  and  its  principal  focus.  By 
Art.  352  (2),  rays  which  diverge  from  a  point  between  the  mirror 
and  its  principal  focus,  continue  to  diverge  after  reflection,  but  in 
a  less  degree.  Let  C  be  the  centre,  and  F  the  principal  focus  of 
the  mirror  M  N  (Fig.  224),  and  A  B  the  object.  Draw  the  axes, 
C  A,  C  B,  and  produce  them  behind  the  mirror.  The  pencil  from 

FIG.  224. 


A  will  be  reflected  to  the  eye  at  H,  radiating  as  from  a,  in  the 
same  axis ;  likewise,  that  from  B,  as  from  b.  Therefore,  the 
image  is  apparent,  since  rays  do  not  actually  flow  from  it;  erect, 
as  the  axes  do  not  cross  each  other  between  the  object  and  image  ; 
enlarged,  because  it  subtends  the  angle  of  the  axes  at  a  greater  dis- 
tance than  the  object  does.  As  the  object  approaches,  and  finally 
reaches  the  principal  focus,  the  reflected  rays  approach  parallelism, 
and  the  image  departs  from  the  mirror,  till  it  is  at  an  infinite  dis- 
tance. 

Other  rays  than  those  given  in  the  figure  fall  upon  the  mirror 
from  A,  but  are  reflected  either  above  or  below  the  eye,  and  there- 
fore have  no  part  in  the  production  of  the  image,  and  for  that 
reason  are  omitted.  The  same  is  true  of  rays  from  every  other 
point  of  the  object. 
16 


240  OPTICS. 

2.  Object  between  the  principal  focus  and  the  centre.  As  soon 
as  the  object  passes  the  principal  focus,  the  rays  of  each  pencil  be- 
gin to  converge  ;  and  each  radiant  of  the  object  has  its  conjugate 
focus  in  the  same  axis  beyond  the  centre  (Art.  353). 

For  example  the  rays  diverging  from  the  point  A,  represented 
in  Fig.  225  by  full  lines,  after  reflection  are  converged  to  a  situ- 
ated somewhere  on  the  secondary  axis  A  C  a,  and  rays  from  B, 

FIG.  235. 


given  as  dotted  lines,  converge  finally  to  b  on  the  axis  B  Cb. 
The  images  of  intermediate  points  are  formed  in  the  same  way. 

If  an  observer  is  beyond  a  b,  the  rays,  after  crossing  at  the 
image,  will  reach  him,  as  though  they  originated  in  a  b  ;  or  if  a 
screen  is  placed  at  a  b,  the  light  which  is  collected  in  the  focal 
points  will  be  thrown  in  all  directions  by  radiant  reflection  from 
the  screen.  Hence,  the  image  is  real ;  it  is  also  inverted,  because 
the  axes  cross  betwsen  the  conjugate  foci ;  and  it  is  enlarged,  since 
it  subtends  the  angle  of  the  axes  at  a  greater  distance  than  the 
object  does.  That  b  C  is  greater  than  B  C,  is  proved  by  joining 
C  G,  which  bisects  the  angle  B  G  b,  and  therefore  divides  B  b  so 
that  B  C  :  Cb::  B  G:  G  b.  As  0  b  is  greater  than  B  G,  so  C  b 
is  greater  than  B  C.  When  the  object  reaches  the  centre,  the 
image  is  there  also,  but  inverted  in  position,  since  rays  which  pro- 
ceed from  one  side  of  C,  are  reflected  to  the  other  side  of  it. 

3.  Object  beyond  the  centre.  This  is  the  reverse  of  (2),  the 
conjugate  foci  having  changed  places ;  a  b,  therefore,  being  the 
object,  A  B  is  its  image,  real,  inverted,  diminished.  As  the  ob- 
ject removes  to  infinity,  the  image  proceeds  only  to  the  principal 
focus  F. 

363.  Illustrated  by  Experiment. — These  cases  are  shown 
/  experimentally  by  placing  a  lamp  close  to  the  mirror,  and  then 

carrying  it  along  the  axis  to  a  considerable  distance  away.  While 
the  lamp  moves  from  the  mirror  to  the  principal  focus,  its  image 
behind  the  mirror  recedes  from  its  surface  to  infinity ;  we  may 
then  regard  it  as  being  either  at  an  infinite  distance  behind,  or  an 
infinite  distance  in  front,  since  the  rays  of  every  pencil  are  par- 


IMAGES    BY    THE    CONVEX    MIRROR.  241 

allel.  After  the  lamp  passes  the  principal  focus,  the  image  ap- 
pears in  the  air  at  a  great  distance  in  front,  and  of  great  size,  and 
they  both  reach  the  centre  together,  where  they  pass  each  other ; 
and,  as  the  lamp  is  carried  to  great  distances,  the  image,  growing 
less  and  less,  approaches  the  principal  focus,  and  is  there  reduced 
to  its  smallest  size.  The  only  part  of  the  infinite  line  of  the  axis 
before  and  behind,  in  which  no  image  can  appear,  is  the  small 
distance  between  the  mirror  and  its  principal  focus. 

If  a  person  looks  at  himself,  so  long  as  he  is  between  the  mir- 
ror and  the  principal  focus,  he  sees  his  image  behind  the  mirror 
and  enlarged.  But  when  he  is  between  the  principal  focus  and 
centre,  the  image  is  real,  and  behind  him ;  the  converging  rays  of 
the  pencils,  however,  enter  his  eyes,  and  give  an  indistinct  view 
of  his  image  as  if  at  the  mirror.  When  he  reaches  the  centre,  the 
pupil  of  the  eye  is  seen  covering  the  entire  mirror,  because  rays 
from  the  centre  are  perpendicular,  and  return  to  it  from  all  parts 
of  the  surface.  Beyond  the  centre,  he  sees  the  real  image  in  the 
air  before  him,  distinct  and  inverted. 

364.  Images  by  the  Convex  Mirror. — The  convex  mir- 
ror affords  no  variety  of  cases,  because  diverging  rays,  which  fall 
upon  it,  are  made  to  diverge  still  more  by  reflection.  In  Fig.  226 
the  pencil  from  A  is  reflected,  as  if  radiating  from  a  in  the  same 

FIG.  226. 


•axis  A  C,  and  that  from  B,  as  from  b  in  the  axis  B  C ;  and  there 
apparent  radiants  are  always  nearer  the  surface  than  the  middle 
point  between  it  and  C  (Art.  354).  The  image  is  therefore 
apparent ;  it  is  erect,  since  the  axes  do  not  cross  between  the 
object  and  image ;  and  it  is  diminished,  as  it  subtends  the  angle 
of  the  axes  at  a  less  distance  than  the  object. 

As  in  Fig.  224,  rays  from  A  and  B,  which  after  reflection  pass 
above  or  below  the  eye,  have  been  omitted ;  only  that  portion  of 
the  mirror  from  which  rays  are  represented  as  being  reflected  has 
any  part  in  the  formation  of  the  image.  For  eyes  in  other  posi- 
tions other  rays  would  be  used,  still  seeming  to  come  from  the 
•same  image  a  b. 


242  OPTICS. 

365.  Caustics  by  Reflection. — These  are  luminous  curved 
surfaces,  formed  by  the  intersections  of  rays  reflected  from  a  hemi- 
spherical concave  mirror.     The  name  caustic  is  given  from  the 
circumstance  that  heat,  as  well  as  light,  is  concentrated  in  the 

focal  points  which  compose  it.  B  A  D 
(Fig.  227),  represents  a  section  of  the 
mirror,  and  B  F  D  of  the  caustic ;  the 
point  F,  where  all  the  sections  of  the 
caustic  through  the  axis  meet  each 
other,  is  called  the  cusp.  When  the 
incident  rays  are  parallel,  as  in  the 
figure,  ''the  cusp  is  at  the  principal 
focus,  that  is,  the  middle  point  be- 
tween A  and  C.  The  rays  near  the 
axis  R  A,  after  reflection  meet  at  the 
cusp  (Art.  352) ;  but  those  a  little  more 

distant  cross  them,  and  meet  the  axis  a  little  further  toward  A. 
And  the  more  distant  the  incident  ray  from  the  axis,  the  further 
from  the  centre  does  the  reflected  ray  meet  the  axis.  Thus  each 
ray  intersects  all  the  previous  ones,  and  this  series  of  intersections 
constitutes  the  curve,  B  F.  The  curve  is  luminous,  because  it 
consists  of  the  foci  of  the  successive  pencils  reflected  from  the 
arc  A  B. 

If  the  incident  rays,  instead  of  being  parallel,  diverge  from  a 
lamp  near  by,  the  form  of  the  caustic  is  a  little  altered,  and  the 
cusp  is  nearer  the  centre.  This  case  may  be  seen  on  the  surface 
of  milk,  the  light  of  the  lamp  being  reflected  by  the  edge  of  the 
bowl  which  contains  it. 

If  parallel  or  divergent  light  falls  on  a  convex  hemispherical 
mirror,  there  will  be  apparent  caustics  behind  the  mirror ;  that  is, 
the  light  will  be  reflected  as  if  it  radiated  from  points  arranged 
in  such  curves. 

366.  Spherical  Aberration  of  Mirrors. — It  has  already 
been  mentioned  (Art.  351),  that  the  statements  in  this  chapter 
relating  to  focal  points  and  images,  as  produced  by  spherical  mir- 
rors, are  true  only  when  the  mirror  is  a  very  small  part  of  the 
whole  spherical  surface.     In  Art.  365  we  have  seen  the  effect  of 
using  a  large  part  of  the  spherical  surface— viz.,  the  rays  neither 
converge  to,  nor  di  verge  from  a  single  point,  but  a  series  of  points 
arranged  in  a  curve.     This  general  effect  is  called  the  spherical 
aberration  of  a  mirror ;  since  the  deviation  of  the  rays  is  due  to 
the  spherical  curvature.    The  deviation,  as  we  have  seen,  is  quite 
apparent  in  a  hemisphere,  or  any  considerable  portion  of  one  ;  but 
it  exists  in  some  degree  in  any  spherical  mirror,  unless  infinitely 
small  compared  with  the  hemisphere. 


REFRACTION     OF     LIGHT.  248 

But  there  are  curves  which  will  reflect  without  aberration. 
Let  a  concave  mirror  be  ground  to  the  form  of  a  paraboloid,  and 
rays  parallel  to  its  axis  will  be  converged  to  the  focus  without 
aberration.  For,  at  any  point  on  such  a  mirror,  a  line  parallel  to 
the  axis,  and  a  line  drawn  to  the  focus,  make  equal  angles  with 
the  tangent,  and  therefore,  equal  angles  with  the  perpendicular  to 
the  surface.  And  rays,  parallel  to  the  axis  of  a  convex  paraboloid, 
will  diverge  as  if  from  its  focus,  on  the  same  account.  Again,  if  a 
radiant  is  placed  at  the  focus  of  a  concave  parabolic  mirror,  the 
reflected  rays  will  be  parallel  to  the  axis,  and  will  illuminate  at  a 
great  distance  in  that  direction.  Such  a  mirror,  with  a  lamp  in 
its  focus,  is  placed  in  front  of  the  locomotive  engine  to  light  the 
track,  and  has  been  much  used  in  light-houses.  If  a  concave  mir- 
ror is  ellipsoidal,  light  emanating  from  one  focus  is  collected  with- 
out aberration  to  the  other,  because  lines  from  the  foci  to  any  point 
of  the  curve  make  equal  angles  with  the  tangent  at  that  point. 

Since  heat  is  reflected  according  to  the  same  law  as  light,  a 
concave  mirror  is  a  burning-glass.  When  it  faces  the  sun,  the 
light  and  heat  are  both  collected  in  a  small  image  of  the  sun  at 
the  principal  focus.  And,  if  no  heat  were  lost  by  the  reflection, 
the  intensity  at  the  focus  would  be  to  that  of  the  direct  rays,  as 
the  area  of  the  mirror  to  the  area  of  the  sun's  image.  Burning 
mirrors  have  sometimes  been  constructed  on  a  large  scale,  by  giv- 
ing a  concave  arrangement  to  a  great  number  of  plane  mirrors. 


CHAPTER   III. 

REFRACTION     OF     LIGHT. 

367.  Division  of  the  Incident  Beam.— When  light  falls 
on  an  opaque  body,  we  have  noticed  that  it  is  arrested,  and  a 
shadow  formed  beyond.     Of  the  light  thus  arrested,  a  portion  is 
reflected,  and  another  portion  lost,  which  is  said  to  be  absorbed  by 
the  body.     When  light  meets  a  transparent  body,  a  part  is  still 
reflected,  and  a  small  portion  absorbed,  but,  in  general,  the  greater 
part  is  transmitted.     The  ratio  of  intensities  in  the  reflected  and 
transmitted  beams  varies  with  the  angle  of  incidence,  but  little 
being  reflected  at  small  angles  of  incidence,  and  almost  the  whole 
at  angles  near  90°. 

368.  Refraction. — The  transmitted  beam  suffers  important 
changes,  one  of  which  is  a  change  in  direction.     This  change  is 


244 


OPTICS. 


FIG.  2$ 


called  refraction)  and  takes  place  at  the  surface  of  a  new  medium. 
In  Fig.  228,  A  C,  incident  upon  R  8,  the  surface  of  a  different 
medium,  is  turned  at  C  into  another 
line,  as  C  E,  which  is  called  the  re- 
fracted ray.  The  angle  E  C  Q,  be- 
tween the  refracted  ray  and  the  perpen- 
dicular is  called  the  angle  of  refraction  ; 
the  angle  G  C  E,  between  the  direc- 
tions of  the  incident  and  the  refracted 
rays,  is  the  angle  of  deviation. 

It  is  a  general  fact,  to  which  there 
are  but  few  exceptions,  that  a  ray  of 
light  in  passing  out  of  a  rarer  into  a  denser  medium  is  refracted 
toward  the  perpendicular  to  the  surface ;  and  in  passing  out  of  a 
denser  into  a  rarer  medium,  it  is  refracted  from  the  perpendicular. 
But  the  chemical  constitution  of  bodies  sometimes  affects  their 
refracting  power.  Some  inflammable  bodies,  as  sulphur,  amber, 
and  certain  oils,  have  a  great  refracting  power  in  comparison  with 
other  bodies ;  and  in  a  given  instance,  a  ray  of  light  in  passing 
out  of  one  of  these  substances  into  another  of  greater  density  may 
be  turned  from  the  perpendicular  instead  of  toward  it.  In  the 
optical  use  of  the  words,  therefore,  denser  is  understood  to  mean, 
of  greater  refractive  power  ;  and  rarer  signifies,  of  less  refractive 
power.  In  Fig.  228,  the  medium  below  R  8  is  of  greater  refrac- 
tive power  than  that  above. 

Let  A  K  (Fig.  229)  represent  a  straight  rod,  the  lower  end  A 

being  beneath  the  surface 
of  water.  The  rays  which 
diverge  from  the  point  A 
are  bent  from  the  perpen- 
dicular A  B.  The  ray  A  d, 
which,  if  prolonged,  would 
enter  the  eye,  is  by  refrac- 
tion bent  so  as  to  pass  be- 
low, while  the  ray  A  C  de- 
viates at  C  and  enters  the 

eye  as  though  coming  from  A',  thus  giving  to  the  rod  the  bent 
appearance  noticed  in  an  oar  when  in  use. 

In  the  same  manner,  the  bottom  of  a  river  appears  elevated, 
and  diminishes  the  apparent  depth  of  the  stream.  Let  a  small 
object  be  placed  in  the  bottom  of  a  bowl,  and  let  the  eye  be  with- 
drawn till  the  object  is  hidden  from  view  by  the  edge  of  the  bowl. 
If  now  the  bowl  be  filled  up  with  water,  the  object  is  no  longer 
concealed,  for  the  light,  as  it  emerges  from  the  water,  is  bent  away 
from  the  perpendicular,  and  brought  low  enough  to  enter  the  eye. 


FIG.  229. 


(•    * 


LIMIT    OF    TRANSMISSION. 


245 


869.  Law  of  Refraction. — The  law  which  is  found  to  hold 
true  in  all  cases  of  common  refraction  is  this  : 

The  angles  of  incidence  and  refraction  are  on  opposite  sides  of 
the  perpendicular  to  the  surface,  and,  for  any  given  media,  the  sines 
of  the  angles  have  a  constant  ratio  for  all  inclinations. 

For  example,  in  Fig.  230,  if  A  C  is  refracted  to  E,  then  a  0 
will  be  refracted  to  e,  so  that  A  D  :  E  F 
::  a  d  :  ef;  and  if  the  rays  pass  out  in 
a  contrary  direction,  the  ratio  is  also 
constant,  being  the  reciprocal  of  the 
former,  viz.,  E  F  :  A  D  ::  ef  :  a  d. 

This  constant  ratio  is  called  the 
Index  of  Refraction  and  is  found  by 
dividing  the  sine  of  the  angle  of  inci- 
dence by  the  sine  of  the  angle  of  refrac- 
tion. 

A  ray  perpendicular  to  the  surface, 
passing  in  either  direction,  is  not  refracted ;  for,  according  to  the 
law,  if  the  sine  of  one  angle  is  zero,  the  sine  of  the  other  must  be 
zero  also. 

The  following  table  gives  the  indices  of  refraction,  the  ray 
being  supposed  to  pass  from  a  vacuum  into  the  substance  ;  such 
indices  are  termed  absolute  indices  : 


Diamond 2.450 

Carbon  disulphide 1.678 

Oil  of  cassia 1.630 

Flint  glass  (mean) 1.600 

Quartz 1.548 

Canada  Balsam.. .  1.540 


Crown  glass  (mean) 1.530 

Alcohol 1.372 

Water 1.336 

Ice 1.309 

Air 1.000294 


FIG.  231. 


370.  Limit  of  Transmission  from  a  Denser  to  a 
Rarer  Medium. — As  a  consequence  of  the  law  of  refraction, 
there  is  a  limit  beyond  which  a  ray  cannot  escape  from  a  denser 
medium.  Let  A  G  (Fig.  231)  be  the 
ray  incident  upon  the  rarer  medium 
RES.  It  will  be  refracted  from  the 
perpendicular  D  F  into  the  direction 
C  E,  so  that  A  D  is  to  E  F  in  a  constant 
ratio  (Art.  369).  If  the  angle  A  CD  be 
increased,  F  C  E  must  also  increase  till 
at  length  its  sine  equals  C  8. 

Suppose    the    denser  medium  to  be 
water  and  the  rarer  air,  then 

Sine  ,4  CD          1 


246  OPTICS. 

hence,  sine  E  0  F=  1.336  x  sine  A  CD.  IfECFbe  increased 
to  90°,  then  sine  90°  =  1  =  1.336  x  sine  A  G  D,  from  which  we- 
find  sine  A  C  D  =  .7485,  the  angle  corresponding  to  which  is 
48°  28'.  If  the  angle  of  incidence  be  greater  than  48°  28',  its  sine 
would  exceed  .7485,  and  therefore  the  sine  of  the  angle  in  air 
should  exceed  unity,  which  is  impossible.  Hence  it  folloAvs,  that 
whenever  the  angle  of  incidence  is  greater  than  that  at  which  the 
sine  of  the  angle  of  refraction  becomes  equal  to  radius,  the  ray 
cannot  be  refracted  consistently  with  the  constant  ratio  of  the 
sines. 

This  is  proved  also  by  experiment ;  the  emerging  ray  increases 
its  angle  of  refraction  till  it  at  length  ceases  to  pass  out.  Beyond 
that  limit  all  the  incident  rays  are  reflected  from  the  inner  surface 
of  the  denser  medium ;  and  this  reflection  is  more  perfect  than 
any  external  reflection,  and  is  called  total  reflection. 

The  limiting  angle  for  diamond  is  24°  12',  and  its  great  bril- 
liancy, when  properly  cut,  is  due  to  numerous  internal  total  re- 
flections which  cause  the  light  to  emerge  in  different  directions. 

371.  Opacity  of  Mixed  Transparent  Media. — Light  in 
passing  from  a  medium  to  a  different  one,  is  partly  reflected  and 
partly  refracted ;  if  this  be  often  repeated  in  a  mixed  medium  no 
light  is  transmitted.     It  is  the   frequency  of  reflection  at  the 
limiting  surfaces  of  air  and  water  that  renders  foam  opaque.     So- 
also  a   transparent   crystal,  -when   crushed,  becomes  an  opaque 
powder.     If  the  powder  be  wetted  with  a  liquid  having  the  same 
refractive  index  as  the  crystal,  the  reflections  will  be  prevented  and 
transparency  will  result. 

372.  Transmission     through     Parallel    Plane    Sur- 
faces.—Let  8  (Fig.  232)  enter  the  medium  A,  and  represent  the 
emergent  ray  by  8'.     Suppose  the  ray  to  enter  from  a  vacuum, 
and  to  emerge  into  a  vacuum  again,  and  call  the  index  of  refrac- 
tion m.     Then 

sine  a  =  m  x  sine  «',  and 

1    • 
sine  a '  =  -  sine  *"  ; 

multiplying  these  together,  we  have  sine  a  =  sine  a",  whence 
FlG  232  a  =  a",  and  the  emergent  and  incident 

rays  are  parallel.  Suppose  the  ray  flf 
to  enter  a  second  medium  B,  bounded 
by  parallel  faces,  it  will  emerge  parallel 
to  S',  and  therefore  parallel  to  S. 
Hence  if  a  ray  traverse  any  number 
of  media  with  parallel  faces,  these 
media  being  separated  by  vacua  the 


RELATIVE     INDICES     OF     REFRACTION.         247 


FIG.  233. 


linaliy  emergent  ray  will  be  parallel  to  the  first  incident  ray  S. 
If  now  the  spaces  between  the  media  be  diminished,  the  result 
will  not  be  changed,  and  finally  when  the  diminution  reaches  its 
limit  the  faces  of  the  media  will  be  in  contact,  and  we  shall  still 
have  the  incident  and  emergent  rays  parallel. 

373.  Determination  of  Relative  Im 
tion. — When  a  ray  passes  from  a  medium  A  into  another  B 
(Fig.  233),  the  absolute  indices  of  these  being  known  the  relative 
index  may  be  found.  Suppose  the 
media  to  be  bounded  by  parallel  plane 
faces.  Let  m  be  the  absolute  index  of 
A,  and  n  that  of  B.  Denote  the 

,  , .      .    ,        sine  a    , 

relative  index,  -: T  by  i.     Suppose 

sine  a  ,    J 

the  ray  S  to  enter  A  from  a  vacuum, 
then 

sine  a  =  m  x  sine  a' 

sine  a'  —  i  x  sine  a" 

sine  a"  —  -  sine  a  since  the  emergent  ray  S'  is  parallel 
to  the  .incident  ray  S  (Art.  372).  By  multiplying  these  equations 
together,  we  find  i  =  —  ;  hence,  to  find  the  relative  index  of  re- 
traction when  a  ray  passes  from  medium  A  into  medium  B,  divide 
the  absolute  index  of  B  by  that  of  A. 

Suppose  a  ray  to  pass  from  air  into  carbon  disulphide,  then 

'   knowing  which  the  deviation  of  the  ray  for  any 


FIG.  234. 


given  angle  of  incidence  can  be  found. 

The  same  principle  may  be  applied  to  find  the  relative  index 
of  two  substances  whose  relative  indices  with  respect  to  a  third     Y 
are  known. 

374.  Transmission  through  a  Medium  Bounded  by 
Inclined  Planes.— A  medium  bounded  by  inclined  planes  is 
called  a  prism.  The  angle 
included  by  the  planes 
through  which  the  light 
passes  is  called  the  re- 
fracting angle  of  the 
prism,  and  the  planes  are 
deviating  planes. 

Let  A  B  (Fig.  234)  be 
the  incident  ray,  and  C  G 


248  OPTICS. 

the  emergent  ray.  The  total  deviation  will  be  G  D  H  =  d- 
Adopting  the  notation  of  the  figure,  we  have  G  D  H.  =  D  B  C  -f- 
C  B  or  d  =  (i  —  i)  +  (e  — e)  =  i  +  e  —  (i1  -j-  e).  Because 
f  the  perpendiculars  through  B  and  C  we  have  r  =  p,  but  p  =  i' 
-j-  e'  =  r  ;  hence,  d  =  i  -f-  e  —  r  ;  that  is  to  say,  the  total  devia- 
tion is  equal  to  the  sz^m  of  the  angles  of  incidence  and  emergence 
diminished  by  the  refracting  angle  of  the  prism. 

375.  Prism  Used  for  Measuring  Refractive  Power. — 

For  any  given  prism  the  deviation  will  depend  upon  the  angles 
of  incidence  and  emergence. 

If  a  prism  rotate  about  an  axis  parallel  to  its  refracting  edge,  a 
position  of  minimum  deviation  will  be  found  such  that  any  rota- 
tion either  to  right  or  left  will  increase  the  deviation  of  the  ray  ; 
if  now  the  angles  of  incidence  and  emergence  be  measured,  they 
will  be  found  equal. 
From  the  equations 
r  =  i'  +  e' 

d  =  i  +  e  —  r,   by  making  i  =  e,  and  consequently 
i'  =  e',  we  obtain 

*  =  i  (r  +  d),  and  i'  =  £  r  ; 
from  which  we  find  the  relative  index  of  refraction 

_  sine  i  sine  £  (r  +  d) 

sine  i'  ~         sine  \  r 

Thus  having  measured  the  refracting  angle  of  the  prism  and 
the  minimum  deviation  of  the  ray  we  can  at  once  determine 
the  index  of  refraction  of  the  substance  of  which  the  prism  is 
formed. 

If  the  angle  r  be  very  small,  d  will  also  be  small,  and  the  ratio 
of  the  angles  may  be  used  instead  of  the  ratio  of  their  sines,  and 

the  formula  then  becomes  m  =  —    —  =  1-1 — . 

This  is  one  of  the  best  methods  by  which  to  determine  the 
index  of  refraction  of  a  solid,  transparent  substance. 

The  final  deviation  of  the  ray  being  unaffected  by  its  passage 
through  glass  plates  with  parallel  faces,  hollow  prisms  formed  of 
such  plates  may  be  filled  with  a  liquid  whose  index  of  refraction 
is  to  be  determined.  A  tube  whose  end  sections  are  glass  planes 
equally  inclined  to  the  axis  of  the  tube,  may  be  used  to  determine 
the  relative  indices  of  gases  and  air,  and  by  exhausting  the  tube 
to  form  a  vacuum,  the  absolute  indices  may  be  found. 

376.  Light  through  One  Surface.— 

1.  Plane  Surface.  When  parallel  rays  pass  into  another  me- 
dium through  a  plane  surface,  they  remain  parallel.  For  the  per- 


LIGHT    THROUGH    ONE    SURFACE. 


249 


pendiculars  being  parallel,  the  angles  of  incidence  are  equal,  and 
therefore  the  angles  of  refraction  are  equal  also,  and  the  refracted 
rays  parallel.  But  a  pencil  of  diverging  rays  is  made  to  diverge 
less,  when  it  enters  a  denser  medium.  For  the  outer  rays  make 
the  largest  angles  of  incidence,  and  are  therefore  most  refracted 
toward  the  perpendiculars,  and  thus  toward  parallelism  with  each 
other.  And  when  diverging  rays  enter  a  rarer  medium,  they  diverge 
more;  because  the  outside  rays  make  the  largest  angles  of  inci- 
dence, and  therefore  the  largest  angles  of 
refraction,  by  which  means  they  spread 
more  from  each  other. 

The  last  case  is  illustrated  when  we 
look  perpendicularly  into  water,  and  see 
its  depth  apparently  diminished  by  about 
one-fourth  of  the  whole.  Let  A  B  (Fig. 
235)  be  the  surface,  and  C  a  point  at  the 
bottom,  from  which  pencils  come  to  the 
eyes  at  E,  E'.  They  will  appear  to  come 
from  D.  As  the  distance  between  the 
pupils  of  the  eyes  is  less  than  2£  inches, 
the  obliquity  of  the  pencil  C  B  E  will  be 
very  slight.  Let  G  F  be  perpendicular 
to  the  surface  A  B.  

The  angle  C—GBH—  angle  of  in- 
cidence ;  and  A  D  B  =  G  B  E  =  angle  of  refraction.  Now,  in  the 
triangle  B  D  C,  B  G  :  B  D  (: :  A  C  :  A  D  nearly)  : :  sin  D  :  sin  C  : : 
sine  of  refraction  :  sine  of  incidence  : :  1.34  :  1.  Hence  the  apparent 
depth  is  one-fourth  less  than  the  real  depth.  The  apparent  depth 
of  water  may  be  diminished  much  more  than  this  by  looking  into 
it  obliquely. 

2.   Convex  surface  of  the  denser.     A   convex  surface  tends  to 
converge  rays.     Let  C'  (Fig.  236)  be  the  centre  of  convexity,  and 

FIG.  236. 


C'  D,  G'  C,  two  radii  produced.  As  rays  are  bent  toicard  the  per- 
pendiculars in  entering  a  denser  medium,  and  as  the  perpendicu- 
lars themselves  converge  to  C'  the  general  effect  of  such  a  surface 
is  to  produce  convergency.  The  pencil,  A  77,  .4  N,  is  merely 
made  less  divergent,  H  D'  N  A  ;  B  H,  B  N  become  parallel, 


250 


OPTICS. 


H  D',  N  B' ;  D  H,  D  A7,  convergent  to  D' ;  the  parallel  rays,  D  H, 
E  N,  convergent  to  E"  ;  the  convergent  pencil,  D  H,  F  N,  more 
convergent  to  F' ;  but  D  H,  C  N,  which  converge  equally  with 
the  radii,  are  not  changed ;  and  D  H,  G  N,  which  converge  more 
than  the  radii,  converge  less  than  before,  to  G'.  The  two  last 
cases,  which  are  exceptions  to  the  general  effect,  rarely  occur  in 
the  practical  use  of  lenses. 

If  we  trace  in  the  opposite  direction  the  rays,  A,  B\  D',  &c., 
comparing  each  with  D'  D,  we  find,  in  this  case  also,  that  the 
convex  surface  tends  to  converge  the  rays,  by  bending  them  from 
their  respective  perpendiculars. 

3.  Concave  surface  of  the  denser.  A  concave  surface  tends  to 
diverge  rays.  Let  C  C',  CD  (Fig.  237),  be  the  radii  of  concavity 
produced.  As  the  radii  diverge  in  the  direction  in  which  the  light 

FIG.  237. 


mores,  the  rays,  being  bent  toivard  them,  will  generally  be  made 
to  diverge  also.  Hence,  parallel  rays,  B  H,  EN,  are  diverged, 
H  D,  N  E' ;  and  diverging  rays,  B  H,  B  N,  are  diverged  more, 
H  D,  N  B'.  If,  however,  rays  diverge  as  much  as  the  radii,  or 
more,  they  proceed  in  the  same  direction,  or  diverge  less,  a  case 
which  rarely  occurs. 

If  the  rays  are  traced  in  the  opposite  direction,  the  tendency 
in  general  to  produce  divergency  appears  from  the  fact  that  the 
perpendiculars  are  now  converging  lines,  and  the  rays  are  refracted 
from  them. 

377.  Lenses. — A  lens  is  a  transparent  medium  bounded  by 
curved  surfaces  whose  centres  of  curvature  lie  upon  a  normal 
common  to  the  two  surfaces.  If  the  radius  of  curvature  is  made 

FIG.  238. 


infinite,  the  corresponding  surface  becomes  a  plane, 
varieties  are  shown  in  Fig.  238. 


The  usual 


CONVEX    AND    CONCAVE    LENS.  251 

A  double  convex  len-s  (A)  consists  of  two  spherical  segments, 
either  equally  or  unequally  convex,  having  a  common  base. 

A  plano-convex  lens  (B)  is  a  lens  having  one  of  its  sides  convex 
and  the  other  plane,  being  simply  a  segment  of  a  sphere. 

A  double  concave  lens  (C)  is  a  solid  bounded  by  two  concave 
spherical  surfaces,  which  may  be  either  equally  or  unequally  con- 
cave. 

A  plano-concave  lens  (D)  is  a  lens  one  of  whose  surfaces  is 
plane  and  the  other  concave. 

A  meniscus  (E)  is  a  lens  one  of  whose  surfaces  is  convex  and 
the  other  concave,  but  the  concavity  being  less  than  the  convexity, 
it  takes  the  form  of  a  crescent,  and  has  the  effect  of  a  convex  lens 
whose  convexity  is  equal  to  the  difference  between  the  sphericities 
of  the  two  sides. 

A  concavo-convex  lens  (F)  is  a  lens  one  of  whose  surfaces  is  con- 
vex and  the  other  concave,  the  concavity  exceeding  the  convexity, 
and  the  lens  being  therefore  equivalent  to  a  concave  lens  whose 
concavity  is  equal  to  the  difference  between  the  sphericities  of  the 
two  sides. 

A  line  (M  N)  passing  through  a  lens,  perpendicular  to  its  op- 
posite surfaces,  is  called  the  axis.  The  axis  usually,  though  not 
necessarily,  passes  through  the  centre  of  the  figure. 

378.  General   Effect  of  the   Convex   Lens.— Whether 

double-convex  or  plano-convex,  its  general  effect  is  to  converge  light. 
It  has  been  shown  (Art.  376)  that  the  convex  surface  of  a  denser 
medium  tends  to  converge  rays,  whichever  way  they  pass  through 
it.  Therefore,  if  E  (Fig.  239)  is  a  radiant,  while  E  C'  C  follows 


FIG 


the  axis  without  change  of  direction,  the  oblique  ray  E  D  is  first 
refracted  toward  D  C,  and  then  from  C'  D'  produced,  and  both 
actions  conspire  to  converge  it  to  the  axis.  The  rays  are  repre- 
sented as  meeting  in  the  focus  F.  Whether  the  rays  are  actually 
converged,  depends  on  their  previous  relation  to  each  other.  If 
the  lens  is  plano-convex,  the  plane  surface  has  usually  but  little 
effect  in  converging  the  light ;  but  by  Art.  376  it  may  be  shown 
that  its  action  will  usually  conspire  with  that  of  the  convex  sur- 
face. 


252 


OPTICS. 


379.  General  Effect  of  the  Concave  Lens.— This  lens, 
whether  double-concave  or  plano-concave,  tends  to  produce  diver- 
gency. This  is  evident  from  what  has  been  shown  in  Art.  376. 
The  ray  E  D  (Fig.  240),  in  entering  the  denser  medium,  is  first 


FIG.  240. 


FIG.  241. 


refracted  toward  C'  D  produced,  and  on  leaving  the  medium  at  Z)', 
is  refracted  from  D'  C ';  and  is  thus  twice  refracted  from  the  ray 
E  (7,  which  being  in  the  axis,  is  not  refracted  at  all.  If  the  lens 
is  plano-concave,  the  effect  of  the  plane  surface  may,  or  may  not, 
conspire  with  that  of  the  concave  surface. 

380.  The  Optic  Centre  of  a  Lens. — The  incident  and 
emergent  portions  of  a  ray  which  enters  and  leaves  a  lens  at  the 
points  of  contact  of  parallel  tangent  planes  will  be  parallel 
according  to  Art.  372. 

The  point  where  the  part  of  such  ray  included  between  the 
bounding  surfaces  cuts  the  axis  of  the  lens,  or  would  cut  it  if  pro- 
duced, is  called  the  optic  centre. 

In  Fig.  241  let  a  and  b  be  points  of  contact  of  parallel  tangent 

planes,  then  the  radii  C  a 
and  C'  b  being  perpendicu- 
lar to  these  parallel  planes 
are  themselves  parallel, 
hence  the  angles  o  and  o 
are  equal ;  the  angles  at 
P  are  also  equal,  and  henc& 
the  triangles  C  a  P  and 
C'  b  P  are  similar,  and 
C'  P  :  C  P  : :  O'  b  :  C  a. 

Represent  the  thickness 
of  the  lens  x  y,  measured  on  the  axis,  by  t,  and  the  distance  from 
P,  the  optic  centre,  to  the  surface  x  by  e ;  also  make  the  radius 
C  a  =  r  and  C'  b  =  r'.  Substituting  these  values  above  we 
have 

r'  —  e  :  r  —  (t  —  e)  : :  r' :  r, 
from  which  we  obtain 

c  =    r't    =      r' 


CONJUGATE    FOCI.  253 

But  this  value  of  e  is  constant  since  r,  r  and  t  are  constant  \ 
therefore  all  rays  which  suffer  no  deviation  in  passing  through 
the  lens  must  pass  through  a  common  point  P,  called  the  optic 
centre.  The  optic  centre  is  within  the  lens  in  the  cases  of  double 
concave  and  double  convex  lenses,  but  without  in  the  meniscus 
and  concavo-convex.  If  r  =  r'  the  optic  centre  is  midway  between 
the  faces. 

381.  Conjugate  Foci.— If  the  rays  from  R  (Fig.  242)  are 
collected  at  F,  then  rays  emanating  from  F  will  be  returned 
to  R;  and  the  two  points  are  called  conjugate  foci.  Their  relative 
distances  from  the  lens  may  be  determined  when  the  radii  of  the 

FIG.  242. 


surfaces  and  the  index  of  refraction  are  known.  Let  n  be  the 
index  of  refraction,  and  assume,  what  is  practically  true,  that  the 
angles  of  incidence  and  refraction  are  so  small  that  their  ratio  is 
the  same  as  the  ratio  of  their  sines.  Then 

RGP(=KGI)  :IGH::n:I; 

.'.      KG  H-.IGH::  n-l  :1; 
in  like  manner  K  H  G  :  I H  G::n  —  1:1; 

.-.KGH  +  KH  G:I  G  H  +  I H  G:\n-  1:1. 
But      KGH  +  KHG  =  BKF—R+  F-, 
and      IGH+IHG=GIC=C+C'', 
naming  the  acute  angles  at  R,  C,   G'  F,  by  those  letters  re- 
spectively, 

/.  R  +  F :  C  +  O'  ::  n  —  1:1. 

Now,  the  lens  being  thin,  and  the  angles  R,  G,  C",  and  Fvcry 
small,  the  same  perpendicular  to  the  axis,  at  L,  the  centre  of  the 
lens,  may  be  considered  as  subtending  all  those  angles.  Hence, 
each  angle  is  as  the  reciprocal  of  its  distance  from  L.  Let  R  L  = 
p  ;  F  L  =  q;  C  L  =  r ;  and  C"  L  =  r'.  Then  the  equation  above 
becomes, 

S  +  U  +  |::*-l:l; 

p        q    r       r' 

which  expresses  in  general  the  relation  of  the  conjugate  foci. 

382.  To  Find  the  Principal  Focus.— The  radiant  from, 
which  parallel  rays  come  is  at  an  infinite  distance.  Therefore, 


254:  OPTICS. 

making  p  =  <x  ,  and  the  distance  of  the  principal  focus  =  F,  we 
have  -  =  0,  and 


q 

If  the  curvatures  are  equal,  for  crown-glass,  for  which  n  =  -, 

iii 

F  reduces  to  r  ;  that  is,  the  principal  focus  of  a  double  convex 
lens  of  crown-glass,  having  equal  curvatures,  is  at  the  centre  of 
convexity. 

The  foregoing  formulae  are  readily  adapted  to  the  other  forms 
of  lens.     When  a  surface  is  plane,  its  radius  is  infinite,   and 

-,  or  —  =  0.     When  concave,  its  centre  is  thrown  upon  the  same 

side  as  the  surface,  and  its  radius  is  to  be  called  negative.  And  if 
the  focal  distance,  as  given  by  the  formula,  becomes  negative, 
it  is  understood  to  be  on  the  same  side  as  the  radiant  ;  that  is, 
the  focus  is  a  virtual  radiant. 

383.  Powers  of  Lenses  Practically  Determined.  —  The 

iprocal  of  the  principal  focal  length 
power  of  a  lens.     From  Art.  381  we  find 

and  from  Art.  382 
whence  we  have 


reciprocal  of  the  principal  focal  length  of  a  lens  -^,  is  called  the 


As  the  index  of  refraction  and  the  radii  of  curvature  are  not 
generally  known  in  respect  to  any  particular  lens  which  we  may 
happen  to  be  using,  some  practical  method  by  which  to  determine 
F  will  enable  us  to  calculate  readily  either  p  or  q,  the  other  being 
given. 

(1.)  To  find  F  for  a  convex  lens.  —  Form  an  image  of  the  sun 
upon  a  plate  of  ground  glass,  and  measure  the  distance  of  the 
image  from  the  lens.  Or,  place  a  light  on  one  side  of  the  lens 
and  find  its  sharp  image  upon  a  screen  on  the  other  side.  These 

distances  measured,  give  p  and  q,  whence  F  =  . 

(2.)  These  two  methods  assume  the  thickness  of  the  lens  to  be 
email  compared  with  the  focal  length.  The  focal  length  of  a 


EQUIVALENT    COMBINATIONS.  255 

thick  lens,  or  system  of  lenses,  may  be  found  thus :  On  one  side, 
at  a  distance  a  little  greater  than  F}  place  a  scale  strongly  illumi- 
nated by  transmitted  light,  and  receive  the  sharp  and  greatly 
magnified  image  of  one  of  its  divisions  upon  a  screen  upon  the 
other  side  of  the  lens  or  lenses.  Then  let  I  =  length  of  one 
division,  L  =  length  of  its  image,  p  =  distance  of  the  screen 
from  the  lens  (very  great  compared  with  its  thickness),  and  we 

find,  from  similar  right-angled  triangles,  L  :  I : :  p  :  *-=-  =  q,  and 

Jj 

these  values  of  p  and  q  give  .    - 

"~ 


L 

The  focal  length  is  strictly  the  distance  from  F  to  the  intersection 
of  the  axis  by  the  principal  plane  of  the  lens  or  combination  of 
lenses. 

The  principal  plane  passes  through  the  point  of  intersection 
of  an  incident  ray,  parallel  to  the  axis,and  its  emergent  ray,  both 
produced  if  necessary,  and  is  at  right  angles  to  the  axis. 

(3.)  To  find  Ffor  a  concave  lens.  Use  in  contact  with  the 
concave  lens  a  stronger  convex,  of  known  value  for  F,  and  pro- 
ceed according  to  the  preceding  methods.  Then  if  /  =  focal 
length  of  combination  and  /'  =  focal  length  of  convex  alone,  and 
F  =  that  of  concave  lens  sought,  we  shall  find 

-p  —  7.  —  7;, ,  as  will  be  proved  hereafter  ;  or  when  the  lens  is  deep 

and  not  very  small,  take  for  the  focal  length  that  distance  from 
a  screen  at  which  the  circle  of  light  from  the  sun  is  twice  the 
diameter  of  the  lens. 

384.  Equivalent  Combinations.— 

To  find  the  focal  length  of  a  lens  which  shall  be  equal  to  a  com- 
bination of  two  lenses. 

Suppose  the  lenses  (Fig.  243)  to  be  of  such  thickness  as  may 
be  neglected.  Let  a 

ray    parallel    to    the  FIG.  243. 

common  axis  be  inci- 
dent at  R.    If  R  V 
be  drawn  parallel  to 
S   T,    the    emergent 
ray,  A  V  will  repre- 
sent the  focal  length,.  .F,  of  a  lens  which  would  produce  the  same 
deviation  as  this  combination.     Let  A  X  •=.  f  =  focal  length  of 
A,  then  B  X  =  f —  a,  a  being  the  distance  between  B  and  A. 
17 


256  OPTICS. 

Now  if  we  regard  T  as  a  radiant,  and  TSR  as  the  path  of  the 
ray,  then  X  is  the  virtual  conjugate  focus  of  the  lens  B  corre- 
sponding to  T,  and  calling/'  the  focal  length  of  B,  we  have 

Art  383,^  =  -^,-^ 
Substituting  the  value  of  B  X  above,  we  have 


By  similar  triangles  A  V  E,  B  T  S  and  X  A  R,  X  B  S 

BT  B  S      BX 

=  ~r~h  =  -r^?»  whence 


A  V(=F)  ~  AE~  AX 


X 


F-     f          /'(/-a)-        //' 
J.  _  1       1  __  a 
F~f  +  f      //' 
When  the  lenses  are  in  contact  the  distance  a  =  o,  and  we  have 

F  =  ~f  +  J~>>  that  is  to  sav' 

The  power  of  a  combination  of  two  lenses  in  contact  is  equal  to 
the  sum  of  their  respective  powers. 

385.  Images  by  the  Convex  Lens.  —  The  convex  lens  forms 
a  variety  of  images,  whose  character  and  position  depend  on  the 
place  of  the  object.  If  the  position  of  the  object  and  the  focal 
length  of  the  lens  are  known  the  position  of  the  image  can  be 
readily  determined.  Light  reflected  from  any  point  of  the  object 
will  be,  converged  to  the  conjugate  focus  of  that  point.  By  deter- 
mining one  such  conjugate  focus  we  can  determine  the  position 
and  size  of  the  image.  This  one  may  be  found  by  merely  tracing 
the  path  of  two  rays  from  the  same  point  of  the  object  —  one  ray 

parallel  to  the  axis  and 
FIG   244 

the   other   through    the 

optic  centre  of  the  lens. 
The  first  will  be  refracted 
^°  ^ie  Principal  focus  and 
the  latter  will  not  be  de- 
viated. The  point  of  in- 
tersection of  these  rays  is  the  required  conjugate  focus.  In  the 
next  three  diagrams  let  L  be  the  lens,  F  the  principal  focus,  0 
the  object,  and  /  the  image.  In  Fig.  244  let  the  object  be  at  a 
greater  distance  from  the  lens  than  the  focal  length.  The  two  rays 
from  the  head  of  the  object  intersect  at  the  head  of  the  image.  Being 
below  the  axis  shows  that  the  image  is  inverted,  and  being  nearer 
to  the  lens  than  object  shows  that  it  is  smaller.  In  Fig.  245  the 


CAUSTICS     BY     REFRACTION. 


257 


FIG.  245. 


object  is  placed  between  the  principal  focus  and  the  lens.  That  the 
two  rays  may  intersect,  they  must  be  produced  beyond  the  lens.  The 
image  is  then  not 
real  but  apparent, 
i.e.,  rays  from  one 
point  of  the  ob- 
ject do  not  come 
to  a  conjugate  fo- 
cus, but  appear  to 
come  from  the  cor- 
responding point 
of  the  image. 

386.  Images  by  the  Concave  Lens. — The  images  by  con- 
cave lenses  may  be  studied  in  the  same  manner.     In  drawing  the 
ray  which  is  parallel  to  the  axis,  however,  instead  of  converging  it 
to  the  focus  on  the  opposite  side  of  the  lens,  it  must  be  diverged 
from  the  focus  which  is  on  the  same  side  as  the  object.     The  proc- 
ess is  shown  in  Fig.  246. 
Images  from  concave  lenses 
are  always  apparent,  erect 

}•— -— -_____  ,-  -ir/  and  smaller  than  the  ob- 

/!\  It  is  noticeable  that  the 

concave  mirror  and  the  con- 
vex lens  are  analogous  in 

their  effects,  forming  images  on  both  sides,  both  real  and  apparent, 
both  erect  and  inverted,  both  larger  and  smaller  than  the  object  ; 
while  the  convex  mirror  and  the  concave  lens  also  resemble  each 
other,  producing  images  always  on  one  side,  always  apparent,  al- 
ways erect,  always  smaller  than  the  object. 

387.  Caustics  by  Refraction. — If  the  convex  surface  of  a 
lens  is  a  considerable  part  of  a  hemisphere,  the  rays  more  distant 
from  the  axis  will  be  so  much  more  refracted  than  others,  as  to 
•cross  them  and  meet  the  axis  at  nearer 

points,  thus  forming  caustics  by  refrac- 
tion. Fig.  247  shows  this  effect  in  the 
case  of  parallel  rays ;  those  near  the 
axis  intersecting  it  at  the  principal 
focus  F,  and  the  intersections  of  re- 
moter rays  being  nearer  and  nearer  to 
the  lens,  so  that  the  whole  converging 
pencil  assumes  a  form  resembling  a  cone  with  concave  sides. 

The  grating  (Fig.  248),  viewed  through  such  a  lens,  would 


FIG.  247. 


258 


OPTICS. 


appear  distorted,  as  in  Fig.  249,  and  if  viewed  through  a  con- 
cave lens  the  opposite  effect  would  result,  as  in  Fig.  250. 

FIG.  248.  FIG.  249.  FIG.  250. 


388.  Spherical  Aberration  of  a  Lens. — The  production 
of  caustics  is  an  extreme  case  of  what  is  called  spherical  aberra- 
tion.    Unless  the  lens  is  of  small  angular  breadth,  not  more  than 
10°,  a  pencil  whose  rays  originated  in  one  point  of  an  object 
is  not  converged  accurately  to  one  point   of   the  image,  but 
the  outer  rays  are   refracted  too  much,   and  make  their  focus 
nearer  the  lens  than  that  of  the  central  rays,  as  represented  in 
Fig.  251.     If  F  is  the  focus 

of  the  central  rays,  and  F'  FIG.  251. 

of  the  extreme  ones,  other 

rays  of  the  same  beam  are 

collected    in    intermediate 

points,  and  F  F1  is  called 

the   longitudinal   spherical 

aberration;  and   G  H,  the 

breadth  covered  by  the  pencil  at  the  focus  of  central  rays,  is  called 

the  lateral  spherical  aberration. 

Such  a  lens  cannot  form  a  distinct  image  of  any  object ;  be- 
cause perfect  distinctness  requires  that  all  rays  from  any  one  point 
of  the  object  should  be  collected  to  one  point  in  the  image.  If, 
for  example,  the  beam  whose  outside  rays  are  R  A,  R  B,  comes 
from  a  point  of  the  moon's  disc,  that  point  will  not  be  perfectly 
represented  by  F,  because  a  part  of  its  light  covers  the  circle, 
whose  diameter  is  G  H,  thus  overlapping  the  space  representing 
adjacent  points  of  the  moon.  And  if  that  point  had  been  on  the 
edge  of  the  moon's  disc,  F  could  not  be  a  point  of  a  well-defined 
edge  of  the  image,  since  a  part  of  the  light  would  be  spread  over 
the  distance  F  G  outside  of  it,  and  destroy  the  distinctness  of  its 
outline. 

389.  Remedy  for  Spherical  Aberration. — As  spherical 
lenses  refract  too  much  those  rays  which  pass  through  the  outer 
parts,  it  is  obvious  that,  to  destroy  aberration,  a  lens  is  required 
whose  curvature  diminishes*  toward  the  edges.    Accordingly,  forms 
for  ellipsoidal  lenses  have  been  calculated,  which  in  theory  will 
completely  remove  this  species  of  aberration.      But   no   curved 


MIRAGE.  25£ 

solids  can  be  so  accurately  ground  as  those  whose  curvature  is  uni- 
form in  all  planes,  that  is,  the  spherical.  Hence,  in  practice  it  is 
found  better  to  reduce  the  aberration  as  much  as  possible  by  spher- 
ical lenses,  than  to  attempt  an  entire  removal  of  it  by  other  forms 
which  cannot  be  well  made. 

Lenses,  or  combinations  which  are  free  from  spherical  aberra- 
tion, are  said  to  be  Aplanatic.  By  lessening  the  aperture  of  a  lens 
by  a  suitable  diaphragm,  the  aberration  maybe  much  diminished. 

In  a  plano-convex  lens,  whose  plane  surface  is  toward  the  ob- 
ject, the  spherical  aberration  is  4.5  ;  that  is  (Fig.  251),  F  F1  =  4.5 
times  the  thickness  of  the  lens.  But  the  same  lens,  with  its  con- 
vex side  toward  the  object,  is  far  better,  its  aberration  being  only 
1.17.  In  a  double  convex  lens  of  equal  curvatures,  the  aberration 
is  1.67  ;  if  the  radii  of  curvature  are  as  1  :  6,  and  the  most  convex 
side  is  toward  the  object,  the  aberration  is  only  1.07.  By  placing 
two  plano-convex  lenses  near  each  other,  the  aberration  may  be 
still  more  reduced. 

390.  Atmospheric  Refraction. — The  atmosphere  may  be 
regarded  as  a  transparent  spherical  shell,  whose  density  increases 
from  its  upper  surface  to  the  earth.     The  radii  of  the  earth  pro- 
duced are  the  perpendiculars  of  all  the  laminae  of  the  air ;  and 
rays  of  light  coming  from  the  vacuum  beyond,  if  oblique,  are  bent 
gradually  toward  these  perpendiculars ;   and  therefore  heavenly 
bodies  appear  more  elevated  than  they  really  are.    The  greatest 
elevation  by  refraction  takes  place  at  the  horizon,  where  it  is  about 
half  a  degree. 

391.  Mirage. — This  phenomenon,  called  also  looming,  con- 
sists of  the  formation  of  one  or  more  images  of  a  distant  object, 
caused  by  horizontal  strata  of  air  of  very  different  densities.     Ships 
at  sea  are  sometimes  seen  when  beyond  the  horizon,  and  their 
images  occasionally  assume  distorted  forms,  contracted  or  elon- 
gated in  a  vertical  direction.     These  effects  are  generally  ascribed: 
to  extraordinary  refraction  in  horizontal  strata,  whose  difference 
of  density  is  unusually  great.     Bat  many  cases  of  mirage  seem  to 
be  instances  of  total  reflection  from  a  highly  rarefied  stratum  rest- 
ing on  the  earth.     These  occur  frequently  on  extended  sandy 
plains,  as  those  of  Egypt.     When  the  surface  becomes  heated,  dis- 
tant villages,  on  more  elevated  ground,  are  seen  accompanied  by 
their  images  inverted  below  them,  as  in  water.     As  the  traveler 
advances,  what  appeared  to  be  an  expanse  of  water  retires  before 
him.     By  placing  alcohol  upon  water  in  a  glass  vessel,  and  allow- 
ing them  time  to  mingle  a  little  at  their  common  surface,  the 
phenomena  of  mirage  may  be  artificially  represented. 


OPTICS. 


CHAPTEE    IV. 


1 


DECOMPOSITION  AND  DISPERSION  OF  LIGHT. 

392.  The  Prismatic  Spectrum. — Another  change  which 
light  suffers  in  passing  into  a  new  medium,  is  called  decomposi- 
tion, or  the  separation  of  light  into  colors.  For  this  purpose,  the 
glass  prism  is  generally  employed.  It  is  so  mounted  on  a  jointed 
stand,  that  it  can  be  placed  in  any  desired  position  across  the 
beam  from  the  heliostat.  The  beam,  as  already  noticed,  is  bent 
away  from  the  refracting  angle,  both  in  entering  and  leaving  the 
prism,  and  deviates  several  degrees  from  its  former  direction.  If 
the  light  is  admitted  through  a  narrow  aperture,  F  (Fig.  252),  and 

FIG.  252. 


the  axis  of  the  prism  is  placed  parallel  to  the  length  of  the  aper- 
ture, the  light  no  longer  falls,  as  before,  in  a  narrow  line,  L,  but 
is  extended  into  a  band  of  colors,  R  V,  whose  length  is  in  a  plane 
at  right  angles  to  the  axis  of  the  prism.  This  is  called  the  pris- 
matic spectrum1.  Its  colors  are  usually  regarded  as  seven  in  num- 
ber— red,  orange,  yellow,  green,  blue,  indigo,  violet.  The  red 
is  invariably  nearest  to  the  original  direction  of  the  beam,  and 
the  violet  the  most  remote ;  and  it  is  because  the  elements  of 
white  light  are  unequally  refrangible,  that  they  become  separated, 
by  transmission  through  a  refracting  body.  The  spectrum  is 
properly  regarded  as  consisting  of  innumerable  shades  of  color. 
Instead  of  Newton's  division  into  seven  colors,  many  choose  to 
consider  all  the  varieties  of  tint  as  caused  by  the  combination  of 
three  primitive  colors,  red,  yellow,  and  blue,  varying  in  their  pro- 


COLORS. 


261 


portions  throughout  the  entire  spectrum.  The  number  seven,  as 
perhaps  any  other  particular  number,  must  be  regarded  as  arbi- 
trary. 

The  spectrum  contains  rays  of  other  wave  lengths  than  those 
which  affect  the  eye.  The  rays  of  longest  wave  length  are  crowded 
together  at  and  beyond  the  red,  and  here  the  greatest  heat  is  found 
upon  testing  with  a  thermometer. 

The  chemical  or  actinic  rays  of  shortest  wave  lengths,  are 
found  at  and  beyond  the  violet.  These  invisible  rays  differ  from 
those  which  are  visible  only  in  wave  length. 

Light  from  other  sources  is  also  susceptible  of  decomposition 
by  the  prism;  but  the  spectrum,  though  resembling  that  of  the 
sun,  usually  differs  in  the  proportion  of  the  colors. 

393.  The  Individual  Colors  of  the  Spectrum  cannot  be 
Decomposed  by  Refraction. — If  the  spectrum  formed  by  the 
prism  A  be  allowed  to  fall  on  the  screen  E  D  (Fig.  253),  and  one 
color  of  it,  green  for  example,  be  let  through  the  screen,  and 

FIG.  253. 


received  on  a  second  prism,  B,  it  is  still  refracted  as  before,  but 
all  its  rays  remain  together  and  of  the  same  color.  The  same  is 
true  of  every  color  of  the  spectrum.  Therefore,  so  far  as  re- 
frangibility  is  concerned,  all  the  colors  of  the  spectrum  are  alike 

simple. 

394.  Colors  of  the  Spectrum  Recombined.— It  may  be 

shown,  in  several  ways,  that  if  all  the  colors  of  the  spectrum  be 
combined,  they  will  reproduce  white  liglit.  One  method  is  by 
transmitting  the  beam  successively  through  two  prisms  whose 
refracting  angles  are  on  opposite  sides.  By  the  first  prism,  the 
colors  are  separated  at  a  certain  angle  of  deviation,  and  then  fall 
on  the  second,  which  tends  to  produce  the  same  deviation  in  the 
opposite  direction,  by  which  means  all  the  colors  are  brought  upon 
the  same  ground,  and  the  illuminated  spot  is  white  as  if  no  prism 
had  been  interposed.  Or  the  colors  may  be  received  on  a  series  of 
small  plane  mirrors,  which  admit  of  such  adjustment  as  to  reflect 
all  the  beams  upon  one  spot.  Or  finally,  the  several  colors  can, 
by  different  methods,  be  passed  so  rapidly  before  the  eye  that  their 


262  OPTICS. 

visual  impressions  shall  be  united  in  one  ;  in  which  case  the  illu- 
minated surface  appears  white. 

395.  Complementary  Colors. — If  certain  colors  of  the  spec- 
trum are  combined  in  a  compound  color,  and  the  others  in  an- 
other, these  two  are  called  complementary  colors,  because,  when 
united,  they  will  produce  white.     For  example,  if  green,  blue,  and 
yellow  are  combined,  they  will  produce  green,  differing  slightly 
from  that  of  the  spectrum  ;   the   remaining  colors,  red,  orange, 
indigo,  and  violet,  compose  a  kind  of  purple,  unlike  any  color  of 
the  spectrum.     But  these  particular  shades  of  green  and  purple,  if 
mingled,  will  make  perfectly  white  light,  and  are  therefore  com- 
plementary colors. 

Tyndal  gives  these  as  complementary  :  Ked  and  greenish  blue, 
orange  and  cyanogen  blue,  yellow  and  indigo  blue,  greenish  yellow 
and  violet.  / 

396.  Natural  Colors  of  Bodies. — The  colors  which  bodies 
exhibit,  when  seen  in  ordinary  white  light,  are  owing  to  the  fact 
that  they  decompose   light   by  absorbing   or  transmitting   some 
colors  and  reflecting  the  others.     We  say  that  a  body  has  a  certain 
color,  whereas  it  only  reflects  that  color  ;  a  flower  is  called  red, 
because  it  reflects  only  or  principally  red  light ;  another  yellow, 
because  it  reflects  yellow  light,  &c.     A  white  surface  is  one  which 
reflects  all  colors  in  their  due  proportion  ;  and  such  a  surface, 
placed  in  the  spectrum,  assumes  each  color  perfectly,  since  it  is 
capable  of  reflecting  all.     A  substance  which  reflects  no  light,  or 
but  very  little,  is  black.     What  peculiarity  of  constitution  that  is. 
which  causes  a  substance  to  reflect  a  certain  color,  and  to  absorb 
others,  is  unknown. 

Very  few  objects  have  a  color  which  exactly  corresponds  to  any 
color  of  the  spectrum.  This  is  found  to  result  from  the  fact  that 
most  bodies,  while  they  reflect  some  one  color  chiefly,  reflect  the 
others  in  some  degree.  A  red  flower  reflects  the  red  light  abun- 
dantly, and  perhaps  some  rays  of  all  the  other  colors  with  the  red. 
Hence  there  may  be  as  many  shades  of  red  as  there  can  be  differ- 
ent proportions  of  other  colors  intei-mingled  with  it.  The  same 
is  true  of  each  color  of  the  spectrum.  Thus  there  is  an  infinite 
variety  of  tints  in  natural  objects.  These  facts  are  readily  estab- 
lished by  using  the  prism  to  decompose  the  light  which  bodiea 
reflect. 

397.  The  Continuous  Spectrum.— A  spectrum  which  con- 
tains all  the  different  shades  of  color  is  called  a  continuous  spec- 
trum.    It  may  be  obtained  from  the  light  emitted  by  incandescent 
solids  or  liquids.     The  molecules  of  elementary  solids  have  their 


FRAUENHOFER     LINES. 


263 


FIG.  254. 

RED. 


own  rate  of  vibration,  and,  if  allowed  to  vibrate  in  an  unobstructed 
path,  would  communicate  this  rate  to  the  ether.  They,  however, 
do  not  have  an  unobstructed  path,  but  are  continually  colliding 
with  each  other,  and,  as  a  consequence,  communicate  to  the  ether 
vibrations  of  varying  frequencies.  In  a  short  time 
they  furnish  all  the  frequencies  which  are  repre- 
sented by  the  different  parts  of  the  spectrum.  Ow- 
ing to  persistence  of  vision,  the  eye  is  incapable  of 
detecting  the  absence  of  certain  frequencies  at  a 
given  instant. 
I  ==l 

398.  The   Line   Spectrum.— If  the  light, 
emitted  by  the  incandescent  vapor  of  a  chemical 
element,  be  passed  through  a  narrow  aperture  and 
a  prism,  it  will  give  a  line  spectrum,  i.e.,  instead  of 
all  the  colors  a  few  narrow  bright  lines  will  be  seen. 
The  lines  are  parallel  to  the  axis  of  the  prism. 
Every  element  has  its  own  set  of  lines.     A  sodium 
salt  placed  at  the  edge  of  the  colorless  flame  of  a 
Bunsen  burner  will  give  the  flame  a  yellow  shade, 
and  its  spectrum  will  consist  of  two  narrow,  yellow 
lines.     A  lithium  salt  gives  a  red  line.     The  paths 
of  the  molecules,  when  in  the  form  of  vapor,  are 
unobstructed.     Only  definite  rates  of  vibration  are 
communicated  to  the  ether,  giving  definite  colors. 
The  spectrum,  then,  must  be  a  line  spectrum.   When 
an  element  gives  two  or  more  lines,  it  is  reasonable 
to  suppose  that  its  molecules  vibrate  harmonically. 

399.  Frauenhofer  Lines.— If  white  light  be 
passed   through   a  vapor  of  a  chemical  element, 
which,  however,  is  not  incandescent,  those  vibrations 
which  correspond  to  the  frequencies  of  the  vapor 
molecules,  will  be  absorbed  by  the  vapor.    An  exam- 
ination of  the  spectrum  will  reveal  that  it  is  no  longer 
continuous,  but  has  dark  lines  in  the  color  which 
corresponds  to  the  frequencies  of  the  vibrations  of 
which  it  has  been  robbed.     This  phenomena  has  its 
parallel  in  sound.     A  screen  of  wires,  tuned  to  the 
same  pitch  as  an  organ  pipe  and  interposed  between 
the  pipe  and  a  listener,  will  absorb  the  vibrations  of 
the  pipe  so  as  to  make  them  inaudible. 

The  spectrum  oi  the  sun  contains  a  great  number  of  these  dark 
lines,  and  they  are  called  Frauenhofer  lines  after  the  name  of  their 
•discoverer.  Their  appearance  is  roughly  represented  in  Fig.  254. 


264  OPTICS. 

The  positions  of  over  six  thousand  have  been  determined.  The 
non-luminous  vapors  which  surround  the  sun  absorb  the  vibrations 
corresponding  to  these  lines  from  the  mixture  of  vibrations  sent 
out  by  the  incandescent  centre. 

From  what  has  been  said  it  will  be  seen  that  the  spectrum  fur- 
nishes us  with  a  means  of  determining  the  constitution  of  the  vapor 
envelope  of  the  sun  or  of  stars.  We  have  to  map  out  line  spectra 
of  all  the  chemical  elements,  and  then  match  them  into  the  dark 
lines  in  the  solar  spectrum.  Thus  luminous  sodium  vapor  gives- 
two  lines  in  yellow  ;  solar  spectrum  has  two  equally  broad  dark 
lines  at  the  same  place  in  the  yellow  ;  hence  the  sun's  light  passes 
through  sodium  vapor. 

The  spectra  of  the  same  substance  at  widely  different  tem- 
peratures are  often  remarkably  dissimilar.  At  the  very  high 
temperature  near  the  solid  or  liquid  nucleus  of  the  sun,  the 
chemical  elements  themselves  appear  to  be  broken  up  and  re- 
duced to  simpler  forms  of  matter.  This,  together  with  the  fact 
that  there  are  many  coincidences  between  the  lines  of  the  spectra 
of  different  elements,  has  given  rise  to  a  theory  that  all  the  dif- 
ferent chemical  elements  are  but  modifications  of  a  single  kind  of 
matter. 

400.  The  Spectroscope. — This  instrument  is  used  in  exam- 
ining and  comparing  spectra.     It  consists  of  a  round  horizontal 
stand  upon  which  is  placed  one  or  more  prisms.     Projecting  from 
the  table  are  three  telescopes.     One  of  these  contains  an  adjustable 
aperture  to  receive  the  light  to  be  examined.     The  light  traversing 
it  is  refracted  by  the  prism  and  enters  a  second  telescope,  at  the 
end  of  which  is  placed  the  observer's  eye.     The  third  telescope 
contains  a  transparent  scale.     Light  from  an  auxiliary  source,  e.g., 
a  candle,  passes  through  the  scale  and  is  reflected  from  one  surface 
of  the  prism  into  the  observing  telescope.     The  observer  thus  sees 
a  spectrum  and  scale  overlapping  or  adjoining  it. 

401.  Dispersion  of  Light. — Decomposition  of  light  refers 
I  to  the  fact  of  a  separation  of  colors  ;  dispersion,  rather,  to  the  meas- 
ure or  degree  of  that  separation.    The  dispersive  power  of  a  medium 
indicates  the  amount  of  separation  which  it  produces,  compared 
with  the  amount  of  refraction. 

The  deviation  of  the  line  E  is  usually  taken  as  the  deviation  of 
the  beam  regarded  as  a  whole.  The  difference  of  the  deviations 
of  the  lines  A  and  II  is  the  dispersion. 

For  example,  if  a  substance,  in  refracting  a  beam  of  light 
1°  51'  from  its  course,  separates  the  violet  from  the  red  by  4 , 
then  its  dispersive  power  is  T^T  —  .036.  The  following  table- 


ACHROMATISM. 


gives  the  dispersive  power  of  a  few  substances  much  used   in 
optics  : 

Dispersive  power.  Dispersive  power. 

Oil  of  cassia 0.189         Plate-glass 0.032 

Sulphuret  of  carbon 0.130         Sulphuric  acid 0.031 

Oil  of  bitter  almonds 0.079         Alcohol 0.029 

Flint-glass 0.052         Rock-crystal 0.026 

Muriatic  acid 0.043         Blue  sapphire  0.026 

Diamond .  0.038         Fluor-spar 0.022 

Crown-glass 0.036 

The  discovery  that  different  substances  produce  different  de- 
grees of  dispersion,  is  due  to  Dollond,  who  soon  applied  it  to  the 
removal  of  a  serious  difficulty  in  the  construction  of  optical  in- 
struments. 

402.  Chromatic  Aberration  of  Lenses.— This  is  a  devia- 
tion of  light  from  a  focal  point,  occasioned  by  the  different  re- 
frangibility  of  the  colors.     If  the  surface  of  a  lens  be  covered, 
except  a  narrow  ring  near  the  edge,  and  a  sunbeam  be  transmitted 
through  the  ring,  the  chromatic  aberration  becomes  very  apparent ; 
for  the  most  refrangible  color,  violet,  comes  to  its  focus  nearest, 
and  then  the  other  colors  in  order,  the  focus  of  red  being  most 
remote.     Since  the  distinctness  of  an  image  depends  on  the  ac- 
curate meeting  of  rays  of  the  same  pencil  in  one  point,  it  is  clear 
that  discoloration  and  indistinctness  are  caused  by  the  separation 
of  colors. 

403.  Achromatism.— In  order  to  refract  light,  and  still  keep 
the  colors  united,  it  is  necessary  that,  after  the  beam  has  been 
refracted,  and  thus  separated,  a  substance  of  greater  dispersive 
power  should  be  used,  which  may  bring  the  colors  together  again, 
by  refracting  the  beam  only  a  part  of  the  distance  back  to  its 
original  direction.      For  instance,  suppose   two  prisms,  one  of 
crown-glass  and  one  of  flint-glass,  each  ground  to  such  a  refract- 
ing angle  as  to  separate  the  violet  from  the  red  ray  by  4'.     In 
order  for  this,  the  crown-glass,  whose  dispersive  power  is  .036, 

must  refract  the  beam  1°  51'  ;  for  =0-=-^-,  =  .036  ;  and  the  flmt- 

1    51 

glass,  whose  dispersive  power  is  .052,  must  refract  only  1°  17'; 
f°r  10  17;  =  -052.  Place  these  two  prisms  together,  base  to  edge, 

as  in  Fig.  255,  C  being  the  crown-glass  and  F  the  flint-glass. 
Then  C  will  refract  the  beam  b  I,  downward  1°  51',  and  the  violet, 
v,  4'  more  than  the  red,  r ;  F  will  refract  this  decomposed  beam 


266 


OPTICS. 


upward  1°  17',  and  the  violet  4'  more  than  the  red,  which  will  just 
bring  them  together  at  v  r.     Thus  the  colors  are  united  again,  and 

PIG.  255. 


yet  the  beam  is  refracted  downward  1°  51'  —  1°  17'  =  34',  from 
its  original  direction. 

404.  Achromatic  Lens.— If  two  prisms  can  thus  produce 
achromatism,  the  same  may  be  effected  by  lenses ;  for  a  convex 
lens  of  crown-glass  may  converge  the  rays  of  a  pencil,  and  then  a 
concave  lens  of  flint-glass  may  diminish  that  convergency  suffi- 
ciently to  unite  the  colors.  A  lens  thus  constructed  of  two  lenses 
of  different  materials  and  opposite  curvatures,  so  adapted  as  to 
produce  an  image  free  from  chromatic  aberration,  is  called  an 
achromatic  lens.  Fig.  256  shows  such  a  combination.  The  con- 

Fio.  256. 


vex  lens  of  crown-glass  alone  would  gather  the  rays  into  a  series  of 
colored  foci  from  v.to  r;  the  concave  flint-glass  lens  refracts  them 
partly  back  again,  and  collects  all  the  colors  at  one  point,  F. 

405.  Colors  not  Dispersed  Proportionally. — It  is  assumed 
in  the  foregoing  discussion,  that  when  the  red  and  violet  are 
united,  all  the  intermediate  colors  will  be  united  also.  It  is  found 
that  this  is  not  strictly  true,  but  that  different  substances  separate 
two  given  colors  of  the  spectrum  by  intervals  which  have  different 
ratios  to  the  whole  length  of  the  spectrum.  This  departure  from 
a  constant  ratio  in  the  distances  of  the  several  colors,  as  dispersed 
by  different  media,  is  called  the  irrationality  of  dispersion.  In 
consequence  of  it  there  will  exist  some  slight  discoloration  in  the 
Image,  after  uniting  the  extreme  colors.  It  is  found  better  in 
practice  to  fit  the  curvatures  of  the  lenses,  for  uniting  those  rays 
which  most  powerfully  affect  the  eye. 


THE     RAINBOW.  267 

In  a  well-corrected  telescope,  when  pointed  at  a  bright  object, 
such  as  Jupiter  or  the  moon,  a  purple  color  will  be  seen  when  the 
eye-piece  is  pushed  inwards  from  its  position  of  adjustment, 
and  a  greenish  color  will  show  when  the  eye-piece  is  pulled  out 
too  far. 


CHAPTER    Y. 

RAINBOW     AND    HALO. 

406.  The   Rainbow. — This  phenomenon,  when  exhibited 
most  perfectly,  consists  of  two  colored  circular  arches,  projected 
on  falling  rain,  on  which  the  sun  is  shining  from  the  opposite  part 
of  the  heavens.     They  are  called  the  inner  or  primary  bow,  and 
the  outer  or  secondary  bow.     Each  contains  all  the  colors  of  the 
spectrum,  arranged  in  contrary  order  ;  in  the  primary,  red  is  out- 
ermost ;  in  the  secondary,  violet  is  outermost.    The  primary  bow 
is  narrower  and  brighter  than  the  secondary,  and  when  of  unusual 
brightness,  is  accompanied  by  supernumerary  bows,  as  they  are 
called;  that  is,  narrow  red  arches  just  within  it,  or  overlapping 
the  violet ;  sometimes  three  or  four  supernumeraries  can  be  traced 
for  a  short  distance.     The  common  centre  of  the  bows  is  in  a  line 
drawn  from  the  sun  through  the  eye  of  the  spectator. 

407.  Action  of  a  Transparent  Sphere  on  Light. — Let 

a  hollow  sphere  of  glass  be  filled  with  water,  and  cause  a  beam  of 
parallel  rays  of  homogeneous  yellow  light  to  fall  upon  it.  To  pre- 
vent confusion  in  Fig.  257,  we  will 
consider  only  those  rays  which  fall 
upon  the  upper  half  of  the  section 
of  the  sphere,  and  will  trace  them 
as  they  emerge  at  the  lower  half. 
Those  rays  which  enter  near  the 
axis  8'  m  will  be  refracted  to  points 
near  n.  Rays  still  farther  from 
8'  m  will  be  refracted  to  points  still 
farther  from  n.  Rays  at  about  59° 
from  m,  at  A,  will  be  refracted  to 
B,  and  no  ray,  no  matter  where  it 
may  enter  the  sphere,  can  be  re- 
fracted to  a  point  higher  than  A  Now  as  B  is  the  limit  of  the 
arc  n  B,  it  follows  that  rays  close  to  the  middle  ray  of  the  pencil 


268  OPTICS. 

S  A,  both  above  it  and  below  it,  will  be  refracted  to  B,  crowded 
together  as  it  were,  and  after  reflection  a  large  portion  will  emerge 
at  D.  As  on  passing  into  the  sphere  at  A  from  air,  these  converged 
to  B,  so  on  emerging  the  reverse  action  takes  place  at  D,  and  we 
have  a  compact  pencil  of  parallel  rays.  An  eye  at  E  would  receive 
an  impression  of  bright  light  in  the  direction  E  D  ;  an  eye  below 
ED  would  receive  no  light  at  all,  and  at  E'  or  E",  while  some 
light  would  be  received  from  the  diverging  rays,  the  impression 
would  be  much  less  vivid  than  at  E.  We  have  been  considering 
only  a  section  of  the  sphere  through  the  axis  ;  if  now  we  conceive 
this  section  to  revolve  about  the  axis  S'  C,  our  beam  S  A  becomes 
a  hollow  cylinder  of  light,  and  the  emergent  beam  E  D  becomes 
an  emergent  hollow  frustum  of  a  cone,  and  if  the  eye  be  placed 
at  any  element  of  this  cone  the  effect  will  be  as  described  for  the 
element  E. 

408.  The  Primary  Bow. — In  the  preceding  article,  homo- 
geneous yellow  light  was  considered.     Let  us  examine  the  results 
when  white  light  from  the 

sun  falls  upon  a  rain-drop.  FIG.  258. 

Suppose  8  A  (Fig.  258)  to 
be  a  beam  of  parallel  rays 
from  the  sun  incident  at 
59°  from  m.  As  B  was  the 
point  at  which  yellow  rays 
of  the  beam  were  concen- 
trated, the  red  rays  which 
are  less  refrangible  will 
all  concentrate  at  R,  the 
distance  R  B  being  very 
greatly  exaggerated  for  the 
sake  of  clearness  in  the  dia- 
gram. After  reflection  these  red  rays  will  emerge  as  a  beam  oi 
parallel  red  rays  at  R.  The  violet  rays  of  the  beam  S  A  being 
most  refrangible  will  all  meet  at  V,  below  B,  and  will  emerge  at 
V  as  a  beam  of  parallel  violet  rays.  Between  these  will  be  beams 
of  the  intermediate  colors  of  the  spectrum.  These  are  beams  of 
parallel  rays,  but  are  not  parallel  beams,  as  is  shown  in  the  figure. 
The  angle  £  included  between  the  incident  beam  8  A  and  the 
emergent  red  beam  produced  backward  to  x  is  found  by  cal- 
culation to  be  42°  2',  and  the  like  angle  for  the  violet  beam  is 
40°  17'. 

409.  Course  of  Rays  in   Secondary  Bow.— If  we  ex- 
amine the  conditions  of  two  internal  reflections  (Fig.  259),  we 
find  that  a  beam  of  monochromatic  light  entering  at  a  certain 


THE     SECONDARY     BOW. 


269 


FIG.  259. 


distance  from  the  axis  S'  n,  about  71°  42",  suffers  the  least  devia- 
tion possible  after  two  reflections,  that  is  to  say,  the  angle  8x1 

will  be  a  minimum.  Bays 
near  this  ray  S  A  of  mini- 
mum deviation  both  on 
the  side  towards  the  axis 
S '  n  and  on  the  side  away 
from  it  will  tend  to  meet 
in  a  focus  at  /  about  £  of 
the  distance  A  B,  and  will 
then  be  reflected  parallel 
to  D,  again  being  reflected 
to  a  focus  at/'  (E  f  =  £ 
E  D),  and  finally  emerg- 
ing at  E,  a  parallel  beam  as  on  entering.  An  observer  at  / 
would  receive  an  intense  beam  of  light  of  the  particular  color 
used. 

Now  substitute  for  the  monochromatic  light,  light  from  the 
sun,  and  the  results  will  be  as  illustrated  in  Fig.  260,  in  which 
the  difference  in  direction  between  the  red  and  the  violet  rays  has 
been  greatly  exaggerated.  At  A  the  red  rays,  following  the  course 
given  in  Fig.  260  are  con- 
verged, cross  at  the  focus, 
and  at  r  are  reflected  as  a 
parallel  beam  to  r' ;  here 
they  are  again  reflected  to 
a  focus,  and  again  diverg- 
ing pass  on  to  r",  where 
they  emerge  as  a  parallel 
beam  r"  R.  The  violet 
rays  are  separated  from 
the  red  at  A,  and  being 
more  refrangible,  take  the  path  indicated  in  the  figure.  The 
angle  A  x  r"  =  50°  59'  and  A  x'  v"  =  54°  9'.  In  order  that  the 
emergent  pencil  may  enter  the  eye  the  incident  ray  must  enter  on 
the  side  of  the  drop  nearest  the  observer.  The  rays  just  con- 
sidered are  the  only  ones  which,  after  two  reflections,  emerge  com- 
pact and  parallel,  and  give  bright  color  at  a  great  distance. 

The  explanation  just  concluded  gives  a  general  and  sufficiently 
exact  conception  of  the  phenomena  of  the  primary  and  secondary 
bows. 

A  rigid  mathematical  analysis  would  take  note  of  the  caustics 
by  reflection  and  refraction,  and  would  vary  slightly  the  places  of 
maximum  illumination.  To  such  analysis,  and  to  the  principle 
of  interference,  the  student  must  refer  for  an  account  of  the  super- 


Fio.  260. 


270 


OPTICS. 


numerary  bows  accompanying  the  primary,  which  are  sometimes 
observed. 

410.  Axis  of  the  Bows.— Let  A  B  D  G  /(Fig.  261)  repre- 
sent the  path  of  the  pencil  of  red  light  in  the  primary  bow.  If 
A  B  and  /  G  are  produced  to  meet  in  K,  the  angle  K  is  the  devi- 
ation, 42°  2',  of  the  incident  and  emergent  red  rays.  Suppose  the 
spectator  at  /,  and  let  a  line  from  the  sun  be  drawn  through  hjs 
position  to  T\  it  is  sensibly  parallel  to  A  B,  and  therefore  the 
angles  I  and  K  are  equal.  As  T  is  opposite  to  the  sun,  the  red 


FIG.  261. 


FIG.  262. 


color  is  seen  at  the  distance  of  42°  2',  on  the  sky,  from  the  point  T\ 
and  so  the  angular  distance  of  each  color  from  T  equals  the  angle 
which  the  ray  of  that  color  makes  with  the  incident  ray.  In  like 
manner,  in  the  secondary  bow,  if  /  T  (Fig.  262)  be  drawn  through 
the  sun  and  the  eye  of  the  observer,  it  is  parallel  to  A  B,  and  the 
angular  distance  of  the  colored  ray  from  T\s  equal  to  K,  the  devi- 
ation of  the  incident  and  emergent  rays.  /  T  is  called  the  axis  of 
the  bows,  for  a  reason  which  is  explained  in  the  next  article. 

411.  Circular  Form  of  the  Bows.— Let  8  0  C  (Fig.  263) 
be  a  straight  line  passing  from  the  sun,  through  the  observer's 

FIG.  263. 


place  at  0,  to  the  opposite  point  of  the  sky ;  and  let  V  0,  R  0  be 
the  extreme  rays,  which  after  one  reflection  bring  colors  to  the  eye 


THE    TWO     BOWS     COMPARED.  271 

at  0,  and  R'  0,  V  0,  those  which  exhibit  colors  after  two  reflec- 
tions ;  then  (according  to  Arts.  408,  409),  V  0  C  =  40°  17',  R  0  C 
=  42°  2',  R'  0  C  =  50°  59',  V  0  C  =  54°  9'.  Now,  if  we  sup- 
pose the  whole  system  of  lines,  8  V  0,  8  V  0,  to  revolve  about 
S  0  C,  as  an  axis,  the  relations  of  the  rays  to  the  drops,  and  to 
each  other,  will  not  be  at  all  changed  :  and  the  same  colors  will 
describe  the  same  lines,  whatever  positions  those  lines  may  occupy 
in  the  revolution.  The  emergent  rays,  therefore,  all  describe  the 
surfaces  of  cones,  whose  common  vertex  is  in  the  eye  at  0 ;  and 
the  colors,  as  seen  on  the  cloud,  are  the  circumferences  of  their 
bases. 

In  a  given  position  of  the  observer,  the  extent  of  the  arches 
depends  on  the  elevation  of  the  sun.  When  on  the  horizon,  the 
bows  are  semicircles;  but  less  as  the  sun  is  higher,  because  their 
centre  is  depressed  as  much  below  the  horizon  as  the  sun  is  ele- 
vated above  it.  From  the  top  of  a  mountain,  the  bows  have  been 
seen  as  almost  entire  circles. 

412.  Colors  of  the  Two  Bows  in  Reversed  Order. — 

Suppose  the  eye  to  receive  a  red  ray  from  a  drop  a  (Fig.  264) ; 

rays  of  all  other  colors  being  more  refrangible  than  the  red  would 

pass  above  the  eye,  as  does  v'.  In  order  that  a  violet  ray  may 
enter  the  eye  it  must  proceed  from  a 
lower  drop,  as  b,  and  the  less  refrangible 
rays  from  this  drop  will  pass  below  the 
eye,  as  at  R'.  Hence  in  the  primary  bow 
the  drops  which  send  violet  to  the  eye 
are  nearer  to  the  axis  of  the  bow  than 
those  which  send  red,  red  being  there- 
fore the  outermost  color.  A  like  exami- 
nation of  the  secondary  bow  shows  that 

red  is  the  innermost  and  violet  the  outermost  color. 

413.  Rainbows,  the  Colored  Borders  of  Illuminated 
Segments  of  the  Sky. — The  primary  bow  is  to  be  regarded  as 
the  outer  edge  of  that  part  of  the  sky  from  which  rays  can  come  to 
the  eye  after  suffering  but  one  reflection  in  drops  of  rain  ;  and  the 
secondary  bow  is  the  inner  edge  of  that  part  from  which  light, 
after  being  twice  reflected,  can  reach  the  eye. 

It  is  found  by  calculation,  that  in  case  of  one  reflection,  the 
incident  and  emergent  rays  can  make  no  inclinations  with  each 
other  greater  than  42°  2'  for  red  light,  and  40°  1?'  for  violet;  but 
the  inclinations  may  be  less  in  any  degree  down  to  0°.  .  There- 
fore, all  light,  once  reflected,  comes  to  the  eye  from  within  the 
primary  bow. 

But  the  angles,  50°  59'  and  54°  9',  are,  by  calculation,  the  least 


272 


OPTICS. 


deviations  of  red  and  violet  light  from  the  incident  rays  after  two 
reflections.  But  the  deviations  may  be  greater  than  these  limits 
up  to  180°.  Therefore  rays  twice  reflected  can  come  to  the  eye 
from  any  part  of  the  sky,  except  between  the  secondary  bow  and 
its  centre. 

It  appears,  then,  that  from  the  zone  lying  between  the  two 
"bows,  no  light,  reflected  by  drops  internally,  either  once  Or  twice, 
can  possibly  reach  the  eye.  Observation  confirms  these  state- 
ments ;  when  the  bows  are  bright,  the  rain  within  the  primary  is 
more  luminous  than  elsewhere ;  and  outside  of  the  secondary  bow, 
there  is  more  illumination  than  between  the  two  bows,  where  the 
cloud,  is  perceptibly  darkest. 

s-— - — "^ 

414.  The  Common  Halo.— This,  as  usually  seen,  is  a  white 
or  colored  circle  of  about  22°  radius,  formed  around  the  sun  or 
moon.     It  might,  without  impropriety,  be  termed  the  frost-bow, 
since  it  is  known  to  be  formed  by  light  refracted  by  crystals  of  ice 
suspended  in  the  air.     It  is  formed  when  the  sun  or  moon  shines 
through  an  atmosphere  somewhat  hazy.    About  the  sun  it  is  a 
white  ring,  with  its  inner  edge  red,  and  somewhat  sharply  defined, 
while  its  outer  edge  is  colorless,  and  gradually  shades  off  into  the 
light  of  the  sky.     Around  the  moon  it  differs  only  in  showing 
little  or  no  color  on  the  inner  edge. 

415.  How  Caused. — The  phenomenon  is  produced  by  light 
passing  through  crystals  of  ice,  having  sides  inclined  to  each  other 
at  an  angle  of  60°.     Let  the  eye  be  at  E  (Fig.  265),  and  the  sun  in 
the  direction  E  S.    Let  S  A,  SB,  &c.,  be  rays  striking  upon  such 
crystals  as  may  happen  to  lie  in  a  position 

to  refract  the  light  toward  S  E  as  an  axis. 
Each  crystal  turns  the  ray  from  the  refract- 
ing edge  on  entering  ;  and  again,  on  leav- 
ing, it  is  bent  still  more,  and  the  emergent 
pencil  is  decomposed.  The  color,  which 
comes  from  each  one  to  the  eye  E,  depends 
on  its  angular  distance  from  E  S,  and 
the  position  of  its  refracting  angle.  The 
angle  of  deviation  for  A  is  E  A  D=S  E  A  ; 
for  JB,  itisSJZ  B,  and  so  on.  It  is  found 
by  calculation,  that  the  least  deviation  for 
red  light  is  21°  45' ;  the  least  for  orange 
must  be  a  little  greater,  because  it  is  a 
little  more  refrangible,  and  so  on  for  the 
colors  in  order.  The  greatest  deviation 
for  the  rays  generally  is  about  43°  13V.  All 
light,  therefore,  which  can  be  transmitted 


FIG.  265. 


THE     MOCK     SUN.  273 

by  such  crystals  must  come  to  the  observer  from  points  some- 
where between  these  two  limits,  21°  45'  and  43°  13'  from  the  sun. 
But  by  far  the  greater  part  of  it,  as  ascertained  by  calculation, 
passes  through  near  the  least  limit.  j 

416.  Its  Circular  Form. — What  takes  place  on  one  side  of 
E  8  may  occur  on  every  side  ;  or,  in  other  words,  we  may  sup- 
pose the  figure  revolved  about  E  S  as  an  axis,  and  then  the  trans- 
mitted light  will  appear  in  a  ring  about  the  sun  S.     The  inner 
«dge  of  the  ring  is  red,  since  that  color  deviates  least ;  just  out- 
side of  the  red  the  orange  mingles  with  it ;  beyond  that  are  the 
red,  orange,  and  yellow  combined ;  and  so  on,  till,  at  the  minimum 
angle  for  violet,  all  the  colors  will  exist  (though  not  in  equal  pro- 
portions), and  the  violet  will  be  scarcely  distinguishable  from 
white.     Beyond  this  narrow  colored  band  the  halo  is  white,  grow- 
ing more  and  more  faint,  so  that  its  outer  limit  is  not  discernible 
.at  all. 

417.  The  Halo,  a  Bright  Border  of  an  Illuminated 
Zone. — As  in  the  rainbow,  so  in  the  halo,  the  visible  band  of 
colors  is  only  the  border  of  a  large  illuminated  space  on  the  sky. 
The  ordinary  halo,  therefore,  is  the  bright  inner  border  of  a  zone, 
which  is  more  than  20°  wide.     The  whole  zone,  except  the  inner 
«dge,  is  too  faint  to  be  generally  noticed,  though  it  is  perceptibly 
more  luminous  than  the  space  between  the  halo  and  the  luminary. 

418.  Frequency  of  the  Halo. — The  halo  is  less  brilliant 
and  beautiful,  but  far  more  frequent,  than  the  rainbow.    Scarcely 
a  week  passes  during  the  whole  year  in  which  the  phenomenon 
does  not  occur.     In  summer  the  crystals  are  three  or  four  miles 
high,  above  the  limit  of  perpetual  frost.     As  the  rainbow  is  some- 
times seen  in  dew-drops  on  the  ground,  so  the  frost-bow,  just  after 
sunrise,  has  been  noticed  in  the  crystals  which  fringe  the  grass. 

419.  The  Mock  Sun. — The  mock  sun,   or  sun-dog,  is  a 
short  arc  of  the  halo,  occasionally  seen  at  22°  distance,  on  the 
right  and  left  of  the  sun,  when  near  the  horizon.     The  crystals, 
which  are  concerned  in  producing  the  mock  sun,  are  supposed  to 
have  the  form  of  spiculce,  or  six-sided  needles,  whose  alternate 
sides  are  inclined  to  each  other  at  an  angle  of  60°  ;  these  falling 
through  the  air  with  their  axes  vertical,  refract  the  light  only  in 
directions  nearly  horizontal,  and  therefore  present  only  the  right 
and  left  sides  of  the  halo. 

In  high  latitudes,  other  and  complex  forms  of  halo  are  fre- 
quent, depending  for  their  formation  on  the  prevalence  of  crys- 
tals of  other  angles  than  60°.  [See  Appendix  for  calculations  of 
the  angular  radius  of  rainbows  and  halo.] 


274 


OPTICS. 


CHAPTER    VI. 

NATURE    OP    LIGHT.— WAVE    THEORY. 

420.  The  Wave  Theory. — Light  has  sometimes  been  re- 
garded as  consisting  of  material  particles  emanating  from  lumi- 
nous bodies.     But  this,  called  the  corpuscular  or  emission  theory, 
has  mostly  yielded  to  the  undulatory  or  wave  theory,  which  sup- 
poses light  to  consist  of  vibrations  in  a  medium.     This  medium, 
called  the  luminiferous  ether,  is  imagined  to  exist  throughout  all 
space,  and  to  be  of  such  rarity  as  to  pervade  all  other  matter. 
It  is  supposed  also  to  be  elastic  in  a  very  high  degree,  so  that 
undulations  excited  in  it  are  transmitted  with  great  velocity. 

There  is  no  independent  evidence  of  the  existence  of  this 
theoretical  ether. 

Eadiant  heat  consists  of  undulations  of  the  same  ether,  which 
differ  from  those  of  light  only  in  being  slower.  For  it  is  a 
familiar  fact,  that  when  the  heat  of  a  body  is  increased  to  about 
500°  or  600°  C.  the  body  becomes  luminous,  and  the  brightness 
increases  as  the  temperature  is  raised. 

421.  Nature  of  the  Wave. — Suppose  a  number  of  ether 
molecules,  as  a,  b,  c,  &c.  (Fig.  266),  to  be  equidistant  upon  the 
straight  line  a  f.     Now  conceive 

that  a  moves  to  «',  then  back  to 
a",  thence  to  a  again,  occupying 
four  equal  intervals  of  time  in 
the  circuit.  Suppose  b  to  start 
on  the  same  round,  at  a  time  one 
interval  later  ;  when  a  reaches 
its  original  position,  and  is  just  beginning  its  upward  motion,  as 
in  the  figure,  b  will  be  at  b"  moving  towards  b.  In  like  manner, 
starting  c  and  d  at  intervals  later  by  one,  we  shall  find  their  posi- 
tions and  directions  of  motion,  when  a  begins  its  second  circuit, 
to  be  as  given  in  the  figure.  The  motion  will  have  been  trans- 
mitted to  e,  which  will  begin  its  first  circuit  at  the  instant  that  a 
starts  upon  its  second.  Molecules  which  like  a  and  e  are  in  the 
same  condition  as  to  place  and  direction  of  motion,  are  said  to  be 
in  the  same  phase.  A  wave  length,  is  the  distance  between 
two  consecutive  like  phases.  The  amplitude  of  vibration  is  the 
distance  between  the  two  limits  of  the  excursion  of  the  particle. 
It  is  evident  from  the  figure  that  the  motion  is  communicated 


THE     WAVE     THEORY.  275 

along  the  axis  a  /  a  distance  a  e,  or  one  wave  length,  during  the 
time  of  one  vibration. 

422.  Postulates  of  the  Wave  Theory.— 

1.  The  loaves  are  propagated  through  the  ether  at  the  rate  of 
300,000,000  metres  (186,300  miles)  per  second. 

As  this  is  the  known  velocity  of  light,  it  must  be  the  rate  at 
which  the  waves  are  transmitted. 

2.  The  atoms  of  the  ether  vibrate  at  right  angles  to  the  line  of 
the  ray  in  all  possible  directions. 

It  was  at  first  assumed  that  the  luminous  vibrations,  like  the 
vibrations  of  sound,  are  longitudinal,  that  is,  back  and  forth  in  the 
line  of  the  ray ;  but  the  discoveries  in  polarization  require  that  the 
vibrations  of  light  shoiild  be  assumed  to  be  transverse,  that  is,  in 
a  plane  perpendicular  to  the  line  of  the  ray  ;  and,  moreover,  that 
in  that  plane  the  vibrations  are  in  every  possible  direction  within 
an  inconceivably  short  space  of  time.  Thus,  if  a  person  is  look- 
ing at  a  star  in  the  zenith,  we  must  consider  each  atom  of  the 
ether  betAveen  the  star  and  his  eye  as  vibrating  across  tho  vertical 
in  all  horizontal  directions,  north  and  south,  east  and  west,  and 
in  innumerable  lines  between  these. 

3.  Different  colors  are  caused  by  different  rates  of  vibration. 
Red  is  caused  by  the  sloiuest  vibrations,  and   violet  by  the 

quickest,  and  other  colors  by  intermediate  rates.  White  light  is 
to  the  eye  what  harmony  is  to  the  ear,  the  resultant  effect  of 
several  rates  of  vibration  combined.  There  are  slower  vibrations 
of  the  ether  than  those  of  red  light,  and  quicker  ones  than  those 
of  violet  light,  but  they  are  not  adapted  to  affect  the  vision.  The 
former  affect  the  sense  of  feeling  as  ft  eat,  the  latter  produce 
chemical  effects,  and  are  called  actinic  rays. 

4.  The  ether  within  bodies  is  less  elastic  than  in  free  space. 

This  is  inferred  from  the  fact  that  light  moves  with  less  veloc- 
ity in  passing  through  bodies  than  in  free  space  ;  the  greater  the 
refractive  power  of  a  body,  the  slower  does  light  move  within  it.' 
And  in  some  bodies  of  crystalline  structure,  it  happens  that  the 
velocity  is  different  in  different  directions,  so  that  the  elasticity  of 
the  ether  within  them  must  be  regarded  as  varying  with  the 
direction, 

423.  Reflection  and  Refraction  according  to  the  Wave 
Theory. — The  vibrations  of  the  ether  are  transmitted  from  the 
source  of  motion  as  spherical  waves.     In  a  luminous  body  are  an 
infinite  number  of  radiants,  each  the  centre  of  a  succession  of 
spherical  waves.    A  beam  of  parallel  rays  is  a  collection  of  parallel 
radii  of  spherical  waves,  having  different  centres  of  disturbance, 
and  the  wave  front  of  such  a  beam  is  the  tangent  plane  common 
to  all  the  spheres. 


276 


OPTICS. 


FIG.  267. 


Suppose  A  and  B  to  be  two  parallel  rays  of  a  beam  of  light. 
Let  a  and  b  (Fig.  267)  represent  two  like  wave  fronts,  and  a  a'  = 
b  b'  be  the  distance  light  moves 
in  any  small  unit  of  time.  When 
the  wave  a  reaches  a',  b  will  have 
reached  b'.  While  b'  moves  to 
6",  a  regarded  as  a  centre  of  dis- 
turbance will  have  sent  out  a 
spherical  wave  to  a".  While  b" 
is  transmitting  a  spherical  wave 
to  b'",  a"  will  have  extended  to 
a",  and  the  common  tangent 
plane  b'"  a"  will  be  the  wave  front, 
and  A'  B',  the  reflected  rays,  rep- 
resent the  beam.  All  rays  from  a'  and  b"  which  move  obliquely 
with  respect  to  each  other  are  destroyed  by  interference,  only 
those  remaining  which  move  in  parallel  directions,  as  A'  B'. 

While  b'  moves  to  b",  a  is  sending  a  wave  into  the  medium  at 
a  slower  rate,  suppose  with  only  half  the  velocity,  which  has  ad- 
vanced to  c  by  the  time  b'  reaches  b".  While  b"  is  moving  into 
the  medium  to  d,  c  has  moved  to  c',  and  a  common  tangent  d  c  is  the 
front  of  the  refracted  beam,  of  which  A"  B"  are  rays. 


FIG.  268. 


424.  Relation  of  Angles  of  Incidence,  Reflection,  and 
Refraction. — The  velocity  of  light  in  any  medium  being  uniform, 

retaining  the  notation  of  Fig. 
267  we  have  in  the  triangles 
a  b'  b"  and  «'  a"  b"  (Fig.  268) 
a'  a"  =;  b'  b",  a'  b"  common, 
and  the  angles  at  a"  and  b' 
equal,  being  right  angles 
formed  by  the  radii  and  tan- 
gents ;  hence  the  angles  a"  a'  b" 
and  b'  b"  a'  are  equal ;  there- 
fore the  incident  and  reflected  rays  muxt  make  equal  angles  with 
the  surface,  and  consequently  with  the  normal. 

In  the  triangle  a'  b"  b'  we  have  b'  b"  =  a'  b"  sin  b'  a  b",  and  in 
the  triangle  a  b''  c  we  have  a'  c  =  a'  b"  sin  a'  b"  c. 

b'  b"  _  sine  b'  a'-b"  _  sine  a?  a'  a •  _     sine  ang.  Inc. 
a'  c        sine  a'  b"  c      sine  c  a!  y~  sine  ang.  Eefrac. 


But  b'  b"  and  a'  c  are  spaces  through  which  the   wave  is  propa- 


INTERFERENCE.  277 

gated  in  the  same  interval  of  time,  and  as  the  velocities  are  con- 
stant in  each  medium,  the  ratio  —, —  must  be  a  constant  ratio ; 

a  c 

,       .       sine  ang.  Inc. 

therefore  its  equal  ratio  s^ne      ~   i|efrac  must  als°  be  a  constant 
ratio. 

In  the  above  it  has  been  assumed  that  the  velocity  of  trans- 
mission in  the  denser  medium  is  less  than  in  the  rarer;  this 
assumption  is  verified  by  direct  experiment. 

425.  Interference.— As  two  systems  of  water-waves  may 
increase  or  diminish  their  height  by  being  combined,  and  as 
sounds,  when  blended,  may  produce  various  results,  and  even 
destroy  each  other,  so  may  two  pencils  of  light  either  augment  or 
diminish  each  other's  brightness,  and  even  produce  darkness. 

If  unlike  phases  coincide,  the  vibrations  are  destroyed  and 
darkness  follows,  while  if  like  phases  meet,  increase  of  brightness 
results. 

To  illustrate  this,  let  two  plane  reflectors,  inclined  at  a  very 
obtuse  angle,  nearly  180°,  receive  light  from  a  minute  radiant,  and 
reflect  it  to  one  spot  on  a  screen ;  the  reflected  pencils  will  inter- 
fere, and  produce  bright  and  dark  lines.  Suppose  light  of  one 
color,  as  violet,  flows  from  a  radiant  point  A  (Fig.  269) ;  let  mir- 
rors B  C  and  B  D  reflect  it  to  the  screen  K  L.  F  and  E  may  be 
so  selected  that  the  ray  A  F  +  F  0  equals  the  ray  A  E  +  E  G. 

FIG.  269. 


Then  G  will  be  luminous,  because  the  two  paths  being  equal,  the 
same  phase  of  wave  in  each  ray  will  occur  at  the  point  G,  But  if 
H  be  so  situated  that  A  f  +  f  H  differs  half  a  violet  wave  from 
A  e  +  e  H,  then  H  will  be  a  dark  point,  because  opposite  phases 
meet  there.  A  similar  point,  /,  will  lie  on  the  other  side  of  G. 
Again,  there  are  two  points,  K  and  L,  one  on  each  side  of  G,  to 
each  of  which  the  whole  path  of  light  by  one  mirror  will  exceed 
the  whole  by  the  other  by  just  one  violet  wave ;  those  points  are 
bright. 

If  the  paths  differ  by  any  odd  multiple  of  |  wave  length,  light 
is  destroyed  and  a  dark  band  is  seen  ;  but  if  these  paths  differ  by 


278  OPTICS. 

any  multiple  of  a  whole  wave  length,  the  light  is  intensified  and 
bright  bands  are  seen. 

Thus,  there  is  a  series  of  bright  and  dark  points  on  the  screen  ; 
or  rather  a  series  of  bright  and  dark  hyperbolic  lines,  of  which 
these  points  are  sections.  Other  colors  will  give  bands  separated 
a  little  further,  indicating  longer  waves.  And  white  light,  produc- 
ing all  these  results  at  once,  will  give  a  repetition  of  the  prismatic 


426.  Striated  Surfaces. — If  the  surface  of  any  substance  is 
ruled  with  fine  parallel  grooves,  2000  or  more  to  the  inch,  these 
grooves  will  act  like  the  inclined  mirrors  of  the  last  paragraph, 
and  it  will  reflect  bright  colors  when  placed  in  the  sunbeam. 
Mother-of-pearl  and  many  kinds  of  sea-shell  exhibit  colors  on 
account  of  delicate  striae  on  their  surface.  It  may  be  known  that 
the  color  arises  from  such  a  cause,  if,  when  the  substance  is  im- 
pressed on  fine  cement,  its  colors  are  communicated  to  the  cement. 
Indeed,  it  was  in  this  way  that  Dr.  Wollaston  accidentally  dis- 
covered the  true  cause  of  such  colors.  The  changeable  hues  in 
the  plumage  of  some  birds,  and  the  wings  of  some  insects,  are  ow- 
ing to  a  striated  structure  of  their  surfaces.  Professor  Eowland 
has  succeeded  in  ruling  43000  lines  to  the  inch  on  speculum  metal, 
this,  too,  on  a  concave  surface. 

Gilt  buttons  and  other  articles  for  dress  are  sometimes  stamped 
with  a  ruling  die,  and  are  called  iris  ornaments.  The  color  in  a 
given  case  depends  on  the  distance  between  the  grooves,  and  the 
obliquity  of  the  beam  of  light.  Hence,  the  same  surface,  uni- 
formly striated,  may  reflect  all  the  colors,  and  every  color  many 
times,  by  a  mere  change  in  its  inclination  to  the  beam  of  light. 

427.  Thin  Laminae. — Thin  films  of  transparent  substances, 
such  as  soap-bubbles,  thin  blown  glass,  oil  on  water,  present  a 
colored  appearance  to  an  observer  viewing  them  by  reflected  or 
transmitted  light.  The  color  varies  with 
changes  in  the  thickness  of  the  film.  The 
phenomena  is  one  of  interference.  To  un- 
derstand it  let  us  consider  A  B  and  A'  B' 
(Fig.  270)  to  be  the  bounding  surfaces  of 
a  thin  film  of  glass,  surrounded  by  air. 
Let  a  ray  of  red  light  come  from  S  and 
strike  the  surface  A  B  at  o.  A  portion  of 
it  will  be  reflected  to  m,  and  the  rest  refracted  to  r,  where  still  an- 
other portion  is  internally  reflected  to  o'  and  then  refracted  to  m'. 
Now,  as  the  film  is  very  thin,  o  m  and  o'  m'  practically  coincide,  i.e., 
o  and  o'  are  coincident,  and  both  rays  would  enter  the  eye  of  an 


NEWTON'S    RINGS.  279 

observer.  At  the  point  r  of  internal  reflection  the  ray  loses  half  a 
wave-length  (Art.  293).  This  always  happens  when  reflection  takes 
place  at  the  surface  of  a  rarer  medium.  Now,  if  the  path  or  o'  (de- 
pendent on  the  thickness  of  the  film)  is  an  even  number  of  half 
icave  lengths  of  red  light,  the  ray  emerging  from  o'  will  be  opposite  in 
phase  to  that  reflected  at  o.  If  the  intensities  of  the  two  rays  were 
the  same,  darkness  would  result  from  the  interference.  For  instance, 
suppose  o  r  o  was  equal  to  four  half  wave  lengths,  i.e.,  two  wave 
lengths ;  adding  the  half  wave  lost  at  r  we  have  light  issuing  from 
o'  two  and  a  half  wave  lengths  behind  o.  It  is  opposite  in  phase  and 
darkness  results.  On  the  other  hand,  if  the  path  o  r  o  is  an  odd 
number  of  half-wave  lengths,  o  and  o'  are  in  the  same  phase,  and 
increase  of  brightness  results.  If  the  observer  moves  his  eye  in 
a  direction  parallel  to  the  surface  of  the  film,  he  sees  brightness 
and  darkness  alternately,  the  obliquity  of  the  rays  altering  the 
length  of  the  path  in  the  film. 

When  S  is  a  source  of  white  light,  colors  result.  For,  in  a 
given  position  of  the' observer,  red  waves  may  be  quenched  while 
others  are  strengthened  in  a  more  or  less  degree. 

If  we  examine  the  transmitted  rays  £71  and  Sr  o'  n',  the  latter  hav- 
ing been  twice  reflected  internally,  we  find,  by  the  same  reasoning, 
that  the  thickness  which  would  produce  opposite  phases  by  reflec- 
tion now  produces  like  phases.  Colors  by  reflection  are  sup- 
planted by  their  complementary  colors  in  transmission. 

428.  Newton's  Rings. — If  a  lens  of  slight  convexity  is  laid 
on  a  plane  lens,  and  the  two  are  pressed  together  by  a  screw,  and 
viewed  by  reflected  light,  rings  of  color  are  seen  arranged  around 
the  point  of  contact.  The  rings  of  least  diameter  are  broadest  and 
most  brilliant,  and  each  one  contains  the  colors  of  the  spectrum  in 
their  order,  from  violet  on  the  inner  edge  to  red  on  the  outer. 
But  the  larger  rings  not  only  become  narrower  and  paler,  but  con- 
tain fewer  colors ;  yet  the  succession  is  always  in  the  same  order 
as  above.  Viewed  by  monochromatic  light  they  appear  as  succes- 
sive rings  of  brightness  and  darkness.  They  are  caused  by  the 
0  interference  between  rays  re- 

FIG.  271.  fleeted  from  the  upper  air  sur- 

face between  the  lenses  (with 
loss  of  half  a  wave  length)  and 
rays  reflected  from  the  lower 
lens.  A  given  color  appears  in 
a  circle  around  the  point  of 

contact,  because  the  circle  marks  the  places  where  the  film  is  of 
equal  thickness.  Measurements  of  the  diameters  of  the  successive 
rings  of  different  colors  show  that  their  squares  are  as  the  odd  num- 


, 


280  OPTICS. 

bers,  1,  3,  5,  7 ;  and  hence  the  thicknesses  of  the  lamina?  of  air  at 
the  repetitions  of  the  same  color  are  as  the  same  numbers.  For,, 
let  Fig.  271  represent  a  section  of  the  spherical  and  plane  surfaces 
in  contact  at  a.  Let  a  b,  a  d,  be  the  radii  of  two  rings  at  their 
brightest  points.  Suppose  a  i,  perpendicular  to  m  n,  to  be  pro- 
duced till  it  meets  the  opposite  point  of  the  circle  of  which  a  g 
is  an  arc,  and  call  that  point  f ;  then  af  is  the  diameter  of  the 
sphere  of  which  the  lens  is  a  segment.  Let  b  e,  d  g,  be  parallel  to 
a  i,  and  e  h,  g  i,  to  m  n,  then  we  have 

(e  h)>  :(giY::ah  x  hf :  a  i  x  if. 

But  the  distances  between  the  two  lenses  being  exceedingly 
small  in  comparison  with  the  diameter  of  the  sphere,  fi/and  if 
may  be  taken  as  equal  to  af;  whence,  by  substitution, 

(e  h)*  :(g  i)*  ::a  h  x  af  :aix  af  : :  a  h  :  a  i  : :  b  e  :  d  g. 
But,  as  has  been  stated,  careful  measurement  of  the  diameters  (or 
radii)  has  shown  that 

(eh?:(giy::l:3. 
Hence  we  have  b  e  :  d  g  : :  1  :  3. 

Therefore  the  thicknesses  of  successive  rings  are  as  the  odd  num- 
bers. 

429.  Thickness  of  Laminae  for  Newton's  Rings. — The 
absolute  thickness,  b  e,  d  g,  &c.,  can  also  be  obtained,  af  being 
known,  since 

af  :a  e  ::  a  e  :a  h  or  b  e ; 

for  in  so  short  arcs  the  chord  may  be  considered  equal  to  the  sine,, 
that  is,  the  radius  of  the  ring. 

With  air  between  the  lenses  Newton  found  the  thickness  of 
the  first  bright  ring  of  orange-yellow  light  to  be  Tnhitnf  °f  an 
inch.  This  distance,  being  twice  traversed  by  one  ray,  and  there 
being  a  loss  of  half  a  wave  at  the  point  of  reflection  of  the  other 
ray,  we  have  -pf-g^^  —  half  a  wave  length  I. 

•'•1  =  TTsW  ==  -000022. 

When  air  is  between  the  lenses,  all  the  rings  range  betv^en 
the  thickness  of  half  a  millionth  of  an  inch  and  72  millionth*;  if 
water  is  used,  the  limits  are  %  of  a  millionth  and  58  millionths. 
Below  the  smaller  limit  the  medium  appears  black,  or  no  color  is 
reflected  ;  above  the  highest  limit  the  medium  appears  white,  all 
colors  being  reflected  together.  When  water  is  substituted  for 
air,  all  the  rings  contract  in  diameter,  indicating  that  a  particular 
order  of  color  requires  less  thickness  of  water  than  of  air ;  the 
thicknesses  for  different  media  are  found  to  be  in  the  inverse  ratio- 
of  the  indices  of  refraction. 


DIFFRACTION. 


281 


430.  Relation  of  Rings  by  Reflection  and  by  Trans- 
mission.— If  the  eye  is  placed  beyond  the  lenses,  the  transmitted 
light  also  is  seen  to  be  arranged  in  very  faint  rings,  the  brightest 
portions  being  at  the  same  thicknesses  as  the  darkest  ones  by 
reflection ;  and  these  thicknesses  are  as  the  even  numbers,  2,  4,  6, 
&c.     The  centre,-  when  black  by  reflection,  is  white  by  transmis- 
sion, and  where  red  appears  on  one  side,  blue  is  seen  on  the  other  ; 
and,  in  like  manner,  each  color  by  reflection  answers  to  its  comple- 
mentary color  by  transmission,  according  to  Article  427. 

431.  Newton's  Rings  by  a  Monochromatic  Lamp. — 

The  number  of  reflected  rings  seen  in  common  light  is  not  usually 
greater  than  from  five  to  ten.  The  number  is  thus  small,  because 
as  the  outer  rings  grow  narrower  by  a  more  rapid  separation  of 
the  surfaces,  the  different  colors  overlap  each  other,  and  produce 
whiteness.  But  if  a  light  of  only  one  color  falls  on  the  lenses,  the 
number  may  be  multiplied  to  several  hundreds ;  the  rings  are 
alternately  of  that  color  and  black,  growing  more  and  more  narrow 
at  greater  distances,  till  they  can  be  traced  only  by  a  microscope. 
A  good  light  for  such  a  purpose  is  the  flame  of  an  alcohol  lamp, 
whose  wick  has  been  soaked  in  strong  brine,  and  dried. 

432.  Diffraction  or  Inflection. — One  of  the  forms  of  inflec- 
tion is  explained  as  follows  :  Through  an  opaque  screen,  A  B 
(Fig.  272),  let  there  be  a  very  narrow  aperture,  c  d,  by  which  is 

admitted  the  beam  of  red 
light,  e  f  g  h,  emanating 
from  a  single  point. 

All  points  of  the  wave 
front  c  d  are  radiants, 
from  which  emanate 
waves  in  all  directions. 
The  original  wave  will 
move  on  through  the 
aperture  forming  the 
band  of  light  y  a  h.  A 
ray  from  c  as  a  radiant  will  interfere  with  that  ray  from  o,  which 
is  one-half  wave  length  in  advance  of  it,  and  will  give  darkness  at 
some  point  j ;  all  rays  parallel  to  c  j  emanating  from  points 
between  c  and  o,  will  interfere  with  corresponding  rays  from 
points  between  o  and  d,  parallel  to  oj,  and  the  total  result  will 
be  the  dark  band  j  Tc. 

In  like  manner  rays  from  those  points  which  differ  by  f  wave 
lengths  will  interfere  and  produce  the  second  dark  bund  m  n. 
Between  these  will  be  a  bright  band,  k  m,  due  to  waves  which 
differ  by  one  whole  wave  length.  The  obliquity  of  the  rays  in  the 


Fia.  272. 


282  OPTICS. 

figure  has  been  very  greatly  exaggerated,  that  the  lines  may  not 
be  confused.  The  general  plan  of  the  determination  of  the  wave 
length  of  the  color  used  may  be  made  plain  by  reference  to  the 
figure.  Let  o  i  be  drawn  perpendicular  to  c  j.  Then  in  the 
triangle  c  o  i  we  have  c  i  =  c  o  x  cos  o  c  i  =  %  c  d  x  sinej  c  g. 

Now.  because  the  screen  is  at  a  great  distance  from  the  slit  as 
compared  with  c  d,  which  is  about  fa  of  an  inch,  and  as  g  cj  is 
very  small  indeed,  about  1.5',  we  nave  ;'  o  =  j  i,  and  hence 
c  j  —  o  j  =  G  j  —  i  j  —  c  i ;  therefore,  in  order  to  find  c  i  (=  \  a 
wave  length)  we  must  measure  c  d  and  the  angular  deviation  g  cj. 
Instead  of  a  single  aperture,  c  d,  a  great  number  of  very  fine 
parallel  lines  ruled  on  glass  are  used,  and  details  of  measurement 
are  adopted,  a  description  of  which  is  beyond  our  limits.  With 
white  light,  prismatic  fringes  would  be  produced. 

433.  Inflection  by  One  Edge  of  an  Opaque  Body. — If 

one  side  of  the  aperture  c  d  in  the  last  paragraph  be  removed,  the 
effect,  while  due  to  the  same  cause,  will  be  somewhat  modified. 
Let  a  convex  lens  converge  sunlight  to  a  focus  from  which  it 
again  diverges,  the  room  being  dark.  If  we  introduce  into  the 
divergent  pencil  any  opaque  body,  as  a  knife-blade,  for  example,  and 
observe  the  shadow  which  it  casts  on  a  white  screen,  we  shall  ob- 
serve on  both  sides  of  the  shadow  fringes  of  colored  light,  the  differ- 
ent colors  succeeding  each  other  in  the  order  of  the  spectrum,  from 
violet  to  red.  Three  or  four  series  can  usually  be  discerned,  the 
one  nearest  to  the  shadow  being  the  most  complete  and  distinct, 
and  the  remoter  ones  having  fewer  and  fainter  colors.  The  phenom- 
enon is  independent  of  the  density  or  thickness  of  the  body  which 
casts  the  shadow.  The  light,  in  passing  by  the  edge  or  back  oi  a 
knife,  by  a  block  of  marble  or  a  bubble  of  air  in  glass,  is  in  each 
case  affected  in  the  same  way.  But  if  the  body  is  very  narrow,  as, 
for  example,  a  fine  wire,  a  modification  arises  from  the  light  which 
passes  the  opposite  side ;  for  now  fringes  appear  within  the 
shadow,  and  at  a  certain  distance  of  the  screen  the  central  line  of 
the  shadow  is  the  most  luminous  part  of  it. 

If  light  of  one  color  be  used,  and  the  distance  of  the  color 
from  the  edge  of  the  shadow  be  measured  when  the  screen  is 
placed  at  different  distances  from  the  body,  it  will  be  found  that 
the  distances  from  the  shadow  are  not  proportional  to  the  distances 
of  the  screen  from  the  body  ;  which  proves  that  the  color  is  not 
propagated  in  a  straight  line,  but  in  a  curve.  These  curves  are 
found  to  be  hyperbolas,  having  their  concavity  on  the  side  next 
the  shadow,  and  are  in  fact  a  species  of  caustics. 

434.  Light  through  Small  Apertures. — The  phenomena 
of  inflection  are  exhibited  in  a  more  interesting  manner  when  we 


SMALL     APERTURES.  283 

view  with  a  magnifying  glass  a  pencil  of  light  after  it  has  passed 
through  a  small  aperture.  For  instance,  in  the  cone  already 
described  as  radiating  from  the  focus  of  a  lens  in  a  dark  room,  let 
a  plate  of  lead  be  interposed,  having  a  pin-hole  pierced  through 
it,  and  let  the  slender  pencil  of  light  which  passes  through  the 
pin-hole  fall  on  the  magnifier.  The  aperture  will  be  seen  as  a 
luminous  circle  surrounded  by  several  rings,  each  consisting  of  a 
prismatic  series.  These  are,  in  truth,  the  fringes  formed  by  the 
edge  of  the  circular  puncture,  but  they  are  modified  by  the  cir- 
cumstance that  the  opposite  edges  are  so  near  to  each  other.  If, 
now,  the  plate  be  removed,  and  another  interposed  having  two 
pin-holes,  within  one-eighth  of  an  inch  of  each  other,  besides  the 
colored  rings  round  each,  there  is  the  additional  phenomenon  of 
long  lines  crossing  the  space  between  the  apertures ;  the  lines  are 
nearly  straight,  and  alternately  luminous  and  dark,  and  varying 
in  color,  according  to  their  distance  from  the  central  one.  These 
lines  are  wholly  due  to  the  overlapping  of  two  pencils  of  light, 
for  on  covering  one  of  the  apertures  they  entirely  disappear. 
By  combining  circular  apertures  and  narrow  slits  in  various 
patterns  in  the  screen  of  lead,  very  brilliant  and  beautiful  effects 
are  produced. 

435.  Why  Inflection  is  not  Always  Noticed  in  Look- 
ing by  the  Edges  of  Bodies. — It  must  be  understood  that 
light  is  always  inflected  when  it  passes  by  the  edges  of  bodies  ; 
but  that  it  is  rarely  observed,  because,  as  light  comes  from  various 
sources  at  once,  the  colors  of  each  pencil  are  overlapped  and  re- 
duced to  whiteness  by  those  of  all  the  others.  By  using  care  to 
admit  into  the  eye  only  isolated  pencils  of  light,  some  cases  of 
inflection  may  be  observed  which  require  no  apparatus.  If  a  per- 
son standing  at  some  distance  from  a  window  holds  close  to  his  eye 
a  book  or  other  object  having  a  straight  edge,  and  passes  it  along 
so  as  to  come  into  apparent  coincidence  with  the  sash-bars  of  the 
window,  he  will  notice,  when  the  edge  of  the  book  and  the  bar 
are  very  nearly  in  a  range,  that  the  latter  is  bordered  with  colors, 
the  violet  extremity  of  the  spectrum  being  on  the  side  of  the  bar 
nearest  to  the  book,  and  the  red  extremity  on  the  other  side. 
Again,  the  effect  produced  when  light  passes  through  a  narrow  aper- 
ture may  be  seen  by  looking  at  a  distant  lamp  through  the  space 
between  the  bars  of  a  pocket-rule,  or  between  any  two  straight 
edges  brought  almost  into  contact.  On  each  side  of  the  lamp  are 
seen  several  images  of  it,  growing  fainter  with  increased  distance, 
and  finely  colored.  An  experiment  still  more  interesting  is  to 
look  at  a  distant  lamp  through  the  net-work  of  a  bird's  feather. 
There  are  several  series  of  colored  images,  having  a  fixed  arrange- 


284:  OPTICS. 

ment  in  relation  to  the  disposition  of  the  minute  apertures  in  the 
feather;  for  the  system  of  images  revolves  just  as  the  feather  itself 
is  revolved. 

436.  Length  and  Number  of  Luminous  "Waves. — The 
careful  measurements  which  have  been  made  in  cases  of  interfer- 
ence have  led,  by  many  independent  methods,  to  the  accurate 
determination  of  the  length  of  a  wave  of  each  color.  When  the 
length  of  a  wave  of  any  color  is  known,  the  number  of  vibrations 
per  second  is  readily  obtained  by  dividing  the  velocity  of  light  by 
the  length  of  the  wave,'  for  light  moves  a  distance  equal  to  one 
wave  length  during  one  vibration  (Art.  421) ;  therefore  if  we  divide 
the  distance  per  second,  300,000,000  meters,  by  the  length  of  one 
wave,  we  have  the  number  of  vibrations  per  second  as  above. 

The  results  of  these  investigations  give  for  the 


Frequencies. 

Wave  lengths  in  cms. 

Ked  (Line  A) 

395  000  000  000  000 

0  00007604 

Yellow  (LineZ)1)  

508,905,810,000,000 

0.00005895 

Violet  (Line  H)       ... 

763,600,000,000,000 

0.00003933 

437.  Calorescence  and  Fluorescence.— Bays  of  less  re- 
frangibility  than  the  extreme  red  are  due  to  vibrations  too  slow  to 
effect  vision,  but  by  their  great  number  they  possess  great  heating 
power.     If  a  beam  of  light  be  allowed  to  fall  upon  a  thin  layer  of 
a  solution  of  iodine  contained  in  a  suitable  cell,  all  light  rays  will 
be  absorbed,  and  nearly  all  the  rays  of  slow  vibration  will  pass 
through.     These,  if  brought  to  a  focus,  will  communicate  to  re- 
fractory substances  vibrations  sufficiently  rapid  to  be  recognized 
by  the  organ  of  vision.     The  phenomenon  is  termed  Calorescence. 

Other  rays,  of  vibrations  too  rapid  to  be  recognized  as  light,, 
are  also  found  in  the  spectrum  far  beyond  the  violet.  If  these  are 
allowed  to  fall  upon  a  solution  of  sulphate  of  quinine,  or  upon 
paper  impregnated  with  sesculine,  or  upon  other  substances  capable 
of  reducing  the  rate  of  vibration,  these  substances  become  visible,. 
glowing  with  a  color  peculiar  to  each  solution,  and  determined 
by  the  rate  of  vibration  which  has  resulted.  This  property  of  sub- 
stances by  which  the  ultra  violet  rays  are  made  visible  is  called 
Fluorescence. 

438.  Phosphorescence. — Very  many  substances,    if    they 
are  exposed  to  a  strong  light  and  then  are  transferred  to  a  dark 
chamber,   continue  to  emit  light  for  longer  or  shorter  periods, 
depending  upon  the  substance  used.     The  sulphides  of  calcium 


POLARIZATION     BY     REFLECTION.  285 

and  strontium  remain  luminous  for  hours  after  exposure  to  sun- 
light. 

Many  other  substances  possess  the  property  of  phosphorescence 
in  so  slight  a  degree,  that  they  will  emit  light  for  only  a  fraction 
of  a  second  after  being  withdrawn  from  the  sun's  rays,  while 
many  seem  not  to  possess  it  at  all. 

What  is  called  phosphorescence  in  certain  animals  and  in 
decaying  animal  and  vegetable  substances  has  no  relation  to  that 
just  described,  and  can  not  properly  be  considered  here. 

I/ 


CHAPTER   VII. 

DOUBLE    REFRACTION    AND    POLARIZATION. 

439.  Change  of  Vibrations  in  Polarized  Light.— It  has 
been  stated  (Art.  422)  that  the  vibrations  of  the  ether  in  the  case 
of  common  light,  must  be  supposed  to  be  transverse  in  all  direc- 
tions.    But,  instead  of  this,  we  may  conceive,  what  is  mechani- 
cally equivalent  to  it,  that  the  vibrations  are  made  in  two  trans- 
verse directions  at  right  angles  to  each  other,  and  to  the  direction 
of  the  ray. 

This  being  the  nature  of  common  light,  it  is  easy  to  state  what 
is  meant  by  polarized  light.  It  is  that  in  which  the  vibrations  are 
performed  in  only  one  of  the  transverse  directions.  It  is,  of 
course,  immaterial  what  particular  transverse  motion  is  cut  off, 
provided  all  the  motion  at  right  angles  to  it  is  retained. 

440.  Polarizing  and  Analyzing  by  Reflection. — When 
light  is  reflected,  those  vibrations  of  the  ray  which  are  in  the- 
plane  of  incidence  are  generally  weakened  in  a  greater  or  less- 
degree,  while  those  which  are  perpendicular  to  the  same  plane  are 
not  affected.     How  much  the  vibrations  are  weakened  depends  on 
the  elasticity  of  the  ether  within  the  medium,  and  on  the  angle 
of  incidence.     But  reflection  of  light  rarely  if  ever  takes  place 
without  diminishing  the  amplitude  of  those  vibrations  which  are 
in  the  plane  of  incidence ;  so  that  a  reflected*  ray  is  always  polar- 
ized, at  least,  in  a  slight  degree. 

441.  Polarization  by  Reflection. — Let  two  tubes,  M  JVaiid 
N  P  (Fig.  273)  be  fitted  together  in  such  a  manner  that  one  can 
be  revolved  upon  the  other ;  and  to  the  end  of  each  let  there  be 
attached  a  plate  of  dark-colored  glass,  A  and  C,  capable  of  reflect- 


-286  OPTICS. 

ing  only  from  the  first  surface.  These  plates  are  hinged  so  as  to 
be  adjusted  at  any  angle  with  the  axis  of  the  tube.  Let  the  plane 
of  each  glass  incline  to  the  axis  of  the  tube  at  an  angle  of  33°, 
and  let  the  beam  R  A  make  an  incidence  of  57°,  the  complement 

Fie.  273. 


of  33°,  on  A  ;  then  it  will,  after  reflection,  pass  along  the  axis  of 
the  tube,  and  make  the  same  angle  of  incidence  on  C.  If  now 
the  tube  N  P  be  revolved,  the  second  reflected  ray  will  vary  its 
intensity,  according  to  the  angle  between  the  two  planes  of  inci- 
dence on  A  and  C.  The  beam  A  C  is  polarized  light ;  the  glass 
A,  which  lias  produced  the  polarization,  is  called  the  polarizing 
j>late  ;  the  glass  C,  which  shows,  by  the  effects  of  its  revolution, 
that  A  C  is  polarized,  is  the  analyzing  plate ;  and  the  whole  in- 
strument, constructed  as  here  represented,  or  in  any  other  manner 
for  the  same  purpose,  is  called  a  polariscope. 

442.  Changes  of  Intensity  Described.— The  changes  in 
the  ray  C  E  are  as  follows  : 

Since  all  vibrations  except  those  perpendicular  to  the  plane  of 
incidence  have  been  destroyed,  when  the  tube  N  P  is  placed  so 
that  the  plane  of  incidence  on  C  is  coincident  with  the  former 
plane  of  incidence,  RAG,  whether  C  E  is  reflected  forward  or 
backward  in  that  plane,  the  intensity  at  E  will  be  the  same  as  if 
A  C  had  been  a  beam  of  common  light.  If  N  P  is  revolved, 
E  will  begin  to  grow  fainter,  and  reach  its  minimum  of  intensity 
when  the  planes  RAG  and  A  C  E  are  at  right  angles,  which  is 
the  position  indicated  in  the  figure  ;  for  only  vibrations  at  right 
angles  to  the  plane  of  incidence  can  be  reflected,  and  there  are  no 
such  vibrations  with  reference  to  this  second  plane  of  incidence. 

Continuing  the  revolution,  we  find  the  intensity  increasing 
through  the  second  quadrant  of  revolution,  and  reaching  its 
maximum,  when  the  two  planes  of  incidence  again  coincide,  180° 
from  the  first  position. 


THE     POLARIZING    ANGLE.  287 

No  reflection  polarizes  perfectly,  and  hence  there  will  be 
increase  and  decrease  of  the  intensity  of  the  reflected  ray,  without 
total  extinction. 

443.  The  Polarizing  Angle.— The  angle  of  57°  is  called 
the  polarizing  angle  for  glass,  not  because  glass  will  not  polarize 
at  other  angles  of  incidence,  but  because  at  all  other  angles  it 
polarizes  the  light  in  a  less  degree ;  and  this  is  indicated  by  the 
fact  that,  in  revolving  the  analyzing  plate,  there  is  less  change  of 
intensity,  and  the  light  at  E  does  not  become  so  faint.  Different, 
substances  have  different  polarizing  angles,  and  for  that  angle  of 
incidence  for  any  substance  which  will  produce  a  maximum  of 
polarization,  the  reflected  and  refracted  rays  will  make  with  each 
other  an  angle  of  90°.  Hence  the  refractive  power  of  opaque 
bodies  may  be  determined.  The  polarization  produced  by  reflec- 
tion from  the  metals  is  very  slight. 

444.  Polarization  by  a  Bundle  of  Plates.  —  Light  may 
also  be  polarized  by  transmission  through  a  bundle  of  lamina?  of 
a  transparent  substance,  at  an  angle  of  incidence  equal  to  its 
polarizing  angle. 

Since  the  reflected  ray  in  perfectly  polarized  light  consists  of 
vibrations  only  at  right  angles  to  the  plane  of  incidence,  the 
transmitted  light,  being  deprived  of  these  vibrations,  will  consist 
of  vibrations  only  in  the  plane  of  incidence  and  refraction.  As 
no  single  reflection  perfectly  sifts  out  the  vibrations  at  right 
angles  to  the  planes  of  incidence  and  reflection,  many  reflections 
at  successive  surfaces  are  secured,  so  that  finally  there  will  remain 
in  the  refracted  ray  only  vibrations  in  the  plane  of  incidence  and 
refraction. 

Let  a  pile  of  twenty  or  thirty  plates  of  transparent  glass,  no 
matter  how  thin,  be  placed  in  the  same  position  as  the  reflector 
A,  in  Fig.  273,  and  a  beam  of  light  be  transmitted  through  them 
in  a  direction  toward  C.  In  entering  and  leaving  the  bundle  A, 
situated  as  in  the  figure,  the  angles  of  incidence  and  refraction 
are  in  a  horizontal  plane.  When  C  is  revolved,  the  beam  under- 
goes the  same  changes  as  before,  with  this  difference,  that  the 
places  of  greatest  and  least  intensity  will  be  reversed.  If  the  light 
is  reflected  from  C  in  the  same  plane  in  which  it  was  refracted  by 
A,  its  intensity  is  least,  and  it  is  greatest  when  reflected  in  a 
plane  at  right  angles  to  it,  as  at  E  in  the  figure. 

445.  Polarization  by  Absorption. — The  third  and  most 
perfect  method  of  polarizing  light,  is  by  transmission  through 
certain  crystals.  Some  crystals  polarize  the  transmitted  light 
by  absorption.  If  a  thin  plate  be  cut  from  a  crystal  of  tour- 


288 


OPTICS. 


FIG.  274. 


maline,  by  planes  parallel  to  its  axis,  the  beam  transmitted  through 
it  is  polarized,  the  vibrations  parallel  to  the  axis  being  transmitted 
and  those  perpendicular  to  the  axis  being  absorbed,  and,  when  re- 
ceived on  the  analyzing ,  plate,  will  alternately  become  bright  and 
faint,  as  the  tube  of  the  analyzer  is  revolved. 

If  the  analyzer  be  a  plate  of  tourmaline  similar  to  the  polarizer 
the  rays  will  pass  when  the  axes  of  the  plates  are  parallel,  but  will 
be  wholly  absorbed  when  the  axes  are  at  right  angles  to  each 
other. 

446.  Double  Refraction. — There  are  many  transparent  sub- 
stances, particularly  those  of  a  crystalline  structure,  which,  instead 
of  refracting  a  beam  of  light  in  the  ordinary  mode,  divide  it  into 
two  beams.     This  effect  is  called  double  refraction, 

and  substances  which  produce  it  are  called  doubly- 
refracting  substances. 

This  phenomenon  was  first  observed  in  a  crys- 
tal of  carbonate  of  lime,  denominated  Iceland  spar. 
It  is  bounded  by  six  rhomboidal  faces,  whose  in- 
clinations to  each  other  are  either  105°  5',  or  74° 
55'.  There  are  two  opposite  solid  angles,  A  and 
X  (Fig.  274),  each  of  which  is  formed  by  the  meeting  of  three  ob- 
tuse plane  angles.  A  line  drawn  in  such  direction  as  to  be  equally 
inclined  to  the  three  edges  of  either  of  these  obtuse  solid  angles, 
is  called  the  axis  of  the  crystal.  If  the  edges  of 
the  crystal  were  equal,  the  axis  would  be  the 
diagonal  A  X  of  the  rhomb. 

A  ray  of  light  passing  through  the  crystal  in 
the  direction  of  the  axis  does  not  suffer  double 
refraction  ;  passing  in  any  other  direction,  how- 
ever, it  does. 

The  elasticity  of  the  ether  in  the  crystal  is 
different  in  the  direction  of  the  axis  from  what 
it  is  in  a  direction  at  right  angles  to  this  axis. 
Now,  remembering  that  the  velocity  of  propaga- 
tion depends  upon  the  elasticity  of  the  medium 
(Art.  422),  and  that  the  velocity  determines  the 
refractive  index  (Art.  424),  we  can  see  that  the  two  sets  of  vibra- 
tions of  a  ray  of  light  R  (Fig.  275),  which  take  place  in  rectangular 
planes,  will  encounter  different  elasticities  upon  entering  a  crystal 
.at  r.  They  will  accordingly  have  different  refractive  indices  and 
will  separate  into  two  rays. 

447.  Ordinary   and    Extraordinary    Rays.  —  Any  plane 
which  contains  the  optical  axis  of  a  crystal  is  termed  a  principal 


FIG.  275. 


KINDS     OF     POLARIZATION.  2S9 

section.  Place  a  principal  section  of  a  crystal  upon  a  black  clot  on 
n  sheet  of  paper.  The  dot  will  appear  as  two,  owing  to  the  double 
refraction.  Kevolve  the  plate  around  an  axis  perpendicular  to  the 
paper.  One  of  the  images  will  remain  stationaiy  while  the  other 
revolves  about  it.  The  rays  which  form  the  stationary  image  will 
be  found  to  conform  to  the  ordinary  laws  of  refraction  (Art.  369), 
nnd  are  termed  the  ordinary  rays.  The  rays  which  form  the  mov- 
nl)le  image  will  be  found  to  depart  from  these  laws.  Either  the 
refractive  index  will  not  be  constant,  or  the  refracted  ray  will  not 
be  in  the  plane  of  incidence.  Both  these  discrepancies  may  occur. 
These  rays  are  called  extraordinary  rays. 

The  refractive  index  of  the  extraordinary  ray  may  be  greater  or 
smaller  than  that  of  the  ordinary  ray,  depending  upon  the  refract- 
ing substance.  Crystals  of  a  positive  axis,  are  those  in  which  the  ex- 
traordinary ray  has  a  larger  index  of  refraction  than  the  ordinary 
ray ;  crystals  of  a  negative  axis  are  those  in  which  the  index  of  the 
extraordinary  ray  is  less  than  that  of  the  ordinary  ray.  Iceland 
spar  is  a  crystal  of  negatiAje  axis. 

Some  crystals  have  two  axes  of  double  refraction ;  that  is,  there 
are  two  directions  in  which  light  may  be  transmitted  without  being 
doubly  refracted.  A  few  crystals  have  more  than  two  axes. 

In  the  experiment  mentioned  above  we  may  assume  that  the 
•crystal  selects  those  vibrations  which  are  in  the  direction  of  the 
optical  axis  to  form  the  ordinary  image,  and  those  at  right  angles 
to  it  to  form  the  extraordinary  image. 

The  property  of  double  refraction  belongs  to  a  large  number  of 
-crystals,  and  also  to  some  animal  substances,  as  hair,  quills,  &c.  ; 
and  it  may  be  produced  artificially  in  glass  by  heat  or  pressure. 

448.  Polarizing  by  Double   Refraction.  —  If  a  beam  is 
passed  through  a  doubly-refracting  crystal,  and  the  two  parts  fall 
on  the  analyzing  plate,  they  will  come  to  their  points  of  greatest 
and  least  brightness  at  alternate  quadrants ;  indeed,  when  one  ray 
is  brightest,  the  other  is  entirely  extinguished.     Therefore  the  two 
rays  which  emerge  from  a  doubly-refracting  crystal  are  polarized 
completely,  the  ordinary  ray  in  a  principal  plane  nnd  the  extraor- 
dinary ray  in  a  plane  at  right  angles  to  a  principal  plane. 

449.  Different  Kinds   of  Polarization. —  Since  the  dis- 
covery was  made  that  the  ethereal  atoms  may  by  certain  methods 
be  thrown  into  circular  movements,  and  by  others  into  vibrations 
in  an  ellipse  with  the  axis  in  a  fixed  direction,  the  polarization 
already  described   has   been  called  plane  polarization,  since  the 
atoms  vibrate  in  a  plane.     Circular  polarization  is  that  in  which 
the  atoms  revolve  in  circles ;  and  elliptical  polarization  denotes  a 


290  OPTICS. 

state  of  vibration  in  ellipses,  whose  major  axes  are  confined  to  one 
plane. 

The  consideration  of  these  various  modes  of  polarization  de- 
mands more  space  than  can  be  spared  here. 

450.  Every  Polarizer  an  Analyzer. — We  have  seen  that 
light  is  polarized  by  reflection  from  glass  at  an  incidence  of  57°, 
and  analyzed  by  another  plate  at  the  same  angle  of  incidence. 
This  is  but  an  instance  of  what  is  always  true,  that  every  method 
of  polarizing  light  may  be  used  to  analyze,  i.e.,  to  test  its  polar- 
ization.    Hence,  a  bundle  61"  thin  plates  of  glass  may  take  the 
place  of  the  analyzer  C,  as  well  as  of  the  polarizer  A.     For,  on 
turning  it  round,  though  the  transmitted  beam  remains  in  the 
same  place,  yet  it  will,  at  the  alternate  quadrants,  brighten  to  its 
maximum  and  fade  to  its  minimum  of  intensity. 

So,  again,  if  light  has  passed  through  a  tourmaline,  and  is 
received  on  a  second  whose  crystalline  axis  is  parallel  to  that  of 
the  former,  the  ray  will  proceed  through  that  also ;  but  if  the 
second  is  turned  in  its  own  plane,  the  transmitted  ray  grows  faint, 
and  nearly  disappears  at  the  moment  when  the  two  axes  are  at  90° 
of  inclination,  and  this  alternation  continues  at  each  90°  of  the 
whole  revolution. 

Finally,  place  a  double-refractor  at  each  end  of  the  polariscope, 
and  let  a  beam  pass  through  them  and  fall  on  a  screen.  The  first 
crystal  will  polarize  each  ray,  and  the  second  will  doubly  refract 
and  also  analyze  each,  exhibiting  a  very  interesting  series  of 
changes.  In  general,  four  rays  will  emerge  from  the  second 
crystal,  producing  four  luminous  spots  on  the  screen.  But,  on 
revolving  the  tube,  not  only  do  the  rays  commence  a  revolution 
round  each  other,  but  two  of  them  increase  in  brightness,  and  the 
other  two  at  the  same  time  diminish  as  fast,  till  two  alone  are 
visible,  at  their  greatest  intensity.  At  the  end  of  the  second 
quadrant,  the  spots  before  invisible  are  at  their  maximum  of 
brightness,  and  the  others  are  extinguished.  This  alternation 
continues  as  long  as  the  crystal  is  revolved.  In  the  middle  of 
each  quadrant  the  four  are  of  equal  brightness. 

The  separation  of  a  plane  polarized  ray  into  two  plane  polarized 
rays  by  a  double  refracting  substance  can  happen  only  when  the 
original  plane  coincides  with  neither  of  the  emergent  planes.  The 
double  refracting,  substance  resolves  the  incident  vibration  into  two 
component  vibrations  at  right  angles  to  each  other.  In  case  the 
original  plane  coincides  with  one  of  the  double  refracting  crystal's 
chosen  planes  the  brightness  in  the  other  plane  will  be  nil. 

451.  Nicol's  Prism. — As  the  four  beams  in  the  last  case  are 


COLOR     BY     POLARIZED     LIGHT. 


291 


FIG.  276. 


an  annoyance  in  investigations  requiring  polarized  light,  only  one 
is  retained,  the  others  being  turned  aside. 

A  rhomb  of  Iceland  spar  is  cut  by  a  plane  A  B  X  passing 
through  the  obtuse  angles.  (Fig.  274) ;   the  two  halves  are  then 

joined  together  again, 
as  they  were  before 
the  cutting,  by  Cana- 
da balsam.  This  is  a 
Nicol  prisrn.  If  a  beam 
enters  the  prism  at  a 
(Fig.  276),  it  will  be 
separated  into  two 
beams,  the  ordinary  and  the  extraordinary,  and  then  will  fall  upon 
the  film  of  Canada  balsam  C  B.  Now  the  refractive  index  of 
Canada  balsam  is  less  than  the  index  of  refraction  for  the  ordinary 
ray,  and  greater  than  that  for  the  extraordinary  ray ;  hence  the 
ordinary  ray  will  be  totally  reflected  at  o,  and  will  pass  out  at  the 
side  of  the  prism,  while  the  extraordinary  ray  will  be  refracted 
through  the  film  of  balsam,  emerging  as  polarized  light  at  m.  If 
a  similar  prisrn  be  used  as  an  analyzer,  one  of  the  two  rays  into 
which  the  polarized  ray  is  separated  is  again  turned  out  of  the 
prism  by  total  reflection,  and  only  a  single  ray  emerges. 

452.  Color  by  Polarized  Light.— Suppose  that  apparatus 
is  arranged  as  in  Fig.  277,  N  being  a  Nicol's  prism  for  a  polarizer, 


L  being  a  lens  which  converges  parallel  rays  from  the  polarizer  to  G, 
which  is  a  section  of  a  double  refracting  crystal  cut  perpendicular 


FIG.  278  (6). 


to  the  optic  axis,  L'  being  another  lens  which  brings  the  rays  to  a 
focus  beyond  the  analyzing  Nicol's  prism  N'.     If,  now,  monochro- 


292  OPTICS. 

matic,  e.g.,  red  light,  be  used,  and  if  the  polarizing  planes  of  N" 
and  N'  be  perpendicular  to  each  other,  the  crystal  being  removed, 
no  light  will  be  perceived  by  an  eye  placed  beyond  N1.  If  the 
crystal  be  now  inserted,  the  eye  sees  a  serifes  of  red  and  black  riugs, 
covered  by  a  black  cross,  as  shown  in  Fig.  278  (a).  Upon  changing 
to  white  light,  the  cross  remains  black,  but  the  rings  assume  pris- 
matic colors.  By  turning  N'  so  that  its  polarizing  plane  coincides 
with  that  of  N,  the  dark  cross  will  change  to  a  white  one  and  the 
colored  rings  will  change  to  their  complementary  colors.  This  is 
indicated  in  Fig.  278  (b). 

In  order  to  explain  these  phenomena,  let  us"  start  with  the 
polarizer  and  analyzer  crossed,  and  with  monochromatic  light.  Any 
vibrations  from  N,  which  are  not  altered  by  the  crystal,  will  fail  to 
pass  through  N'  and  will  give  an  eye  the  impression  of  black.  The 
rays  through  the  optic  axis  are  not  modified,  and  hence  the  black 
centre.  Kays  striking  the  crystal  in  any  other  direction  than  that 
of  the  axis  will  be  doubly  refracted,  splitting  up  into  two  rays  with 
vibrations  in  planes  perpendicular  to  each  other.  One  of  these 
planes  will  pass  through  the  axis  and  the  other  will  be  parallel  to 
the  axis,  but  perpendicular  to  the  first  plane.  The  student  may 
imagine  that  a  uniaxial  doubly  refracting  crystal  is  constructed 
like  the  trunk  of  a  tree.  The  axis  corresponds  to  the  central  heart 
of  the  tree,  and  this  is  surrounded  by  rings  of  the  material.  The 
elasticity  along  the  grain  (and  between  the  rings)  is  different  from 
that  across  the  grain.  Now,  suppose  vibrations  in  a  plane  to 
emanate  from  a  point  on  the  axis  produced.  Those  passing  along 
the  axis  will  be  unaltered.  Those  striking  the  outside  rings  of  the 
crystal  at  points  lying  in  the  plane  of  polarization  passing  through 
the  axis  will  be  singly  refracted,  having  zero  components  in  the 
direction  of  the  tangents  to  the  rings  at  these  points.  The  analyzer 
will  thus  cause  one  arm  of  the  cross  (in  this  plane)  to  be  dark.  In 
the  plane  passing  through  the  axis  and  perpendicular  to  the  one 
just  mentioned,  all  the  vibrations  will  be  in  the  directions  of  the 
tangents,  and  the  component  along  the  radius  will  be  zero.  The 
analyzer  will  thus  cause  the  other  arm  of  the  cross  to  be  dark.  At 
all  other  points  the  vibrations  will  have  definite  components  both 
in  the  tangental  and  the  radial  directions.  The  two  resulting 
rays  will  travel  with  different  velocities,  because  one  has  to  vibrate 
across  the  grain  toward  the  centre  and  the  other  vibrates  with  the 
grain  between  the  concentric  layers.  They  may  thus  emerge  from 
the  crystal  in  different  phases.  Whether  they  are  in  like  or  oppo- 
site phases  depends  upon  the  distance  they  have  had  to  travel  in 
the  crystal.  This  distance  depends  upon  the  obliquity  with  which 
they  strike  the  crystal,  i.e.,  upon  the  distance  of  the  incident  point 


ROTATING  POWER  OF  SUGAR.         293 

from  the  central  axis.  From  equal  distances  from  the  centre,  i.e., 
from  the  circumference  of  a  circle,  they  will  emerge  in  like  phase  ; 
from  another  circle  they  will  emerge  in  opposite  phase.  The  ether 
particles  on  the  emergent  side  of  the  crystal  will  move  under  the 
influence  of  both  systems  of  vibrations.  They  will  desciibe  result- 
ant paths.  These  paths  will  be  in  the  plane  of  N"s  polarization  or 
perpendicular  to  it,  according  as  the  component  vibrations  are  in 
like  or  opposite  phase.  The  impressions  on  the  eye  of  an  observer 
will  be  red  or  black  respectively. 

The  composition  of  the  component  vibrations  and  the  function 
of  the  analyzer  can  be  seen  in  Fig.  279.  Let  the  Figure  represent 
the  surface  of  the  crystal  towards  JV'.  Let  the  inner  circle  rep- 
resent the  locus  of  points  where  the  emergent  vibrations  are  in  like 
phase  and  the  outer  circle  the  locus  of 
those  in  opposite  phase.  N'  N'  is  the 
plane  of  polarization  of  the  analyzer, 
NN  that  of  the  polarizer.  An  ether 
particle  on  the  inner  circle  45°  from 
j^-ATwill  be  subjected  to  two  impulses 
E  and  E',  perpendicular  to  each  other. 
These  will  be  equal  to  each  other,  for 
at  this  angle  a  ray  from  the  polarizer 
is  divided  into  two  equally  intense 
rays.  The  resulting  path  of  the  par- 
ticle will  be  the  diagonal  of  the  parallel- 
ogram. This  corresponds  with  the  analyzer's  plane  of  polarization 
and  the  ray  passes  through  unextinguished.  On  the  outer  circle 
the  direction  of  one  vibration  must  be  changed,  for  here  the  vibra- 
tions are  in  unlike  phase.  Eesolving,  we  get  a  vibration  at  right 
angles  to  the  plane  of  the  analyzer.  The  ray  will,  therefore,  be 
extinguished  by  the  analyzer. 

It  can  be  readily  seen  that  white  light  will  cause  prismatically 
colored  rings  (the  cross  remaining  black),  as  was  explained  in 
Art.  427. 

Making  the  polarization  planes  of  N  and  N'  coincident  will,  of 
course,  change  the  black  cross  to  a  white  one,  and  will  change 
colors  into  their  complementaries. 

453.  Rotation  of  the  Plane  of  Polarization. — Some  sub- 
stances have  the  property  of  twisting  the  plane  of  polarization  of 
light  while  it  is  traversing  them.  For  instance,  if  a  solution  of 
sugar  be  inserted  between  an  analyzer  and  polarizer,  the  former 
of  which  has  been  previously  placed  so  as  to  extinguish  the  light 
from  the  polarizer,  the  light  will  reappear.  In  order  to  again  pro- 
duce extinction,  the  analyzer  must  be  rotated  through  a  certain 


$594  OPTICS. 

angle.  For  a  given  sugar  the  magnitude  of  this  angle  varies  as 
the  concentration  of  the  solution  and  as  the  length  of  it  traversed 
by  the  light.  Some  substances  turn  the  plane  to  the  right,  others 
to  the  left.  This  peculiarity  of  sugar  solutions  is  made  use  of  by 
the  Custom-house  Department  in  appraising  the  value  of  syrups, 
etc. 


CHAPTER    VIII. 

VISION. 

454.  Image  by  Light  through  an  Aperture.— If  light 
from  an  external  object  pass  through  a  small  opening  of  any  shape 
in  the  wall  of  a  dark  room,  it  will  form  an  ill-defined  inverted 
image  on   the  opposite  wall.     Imagine  a  minute  square  orifice, 
through  which  the  light  enters  and  falls  on  a  screen  several  feet 
distant.     A  pencil  of  light,  in  the  shape  of  a  square  pyramid, 
emanating  from  the  highest  point  of  the  object,  passes  through 
the  aperture,  and  forms  a  luminous  square  near  the  bottom  of  the 
screen.     From  an  adjacent  point  another  pencil,  crossing  the  first 
at  the  aperture,   forms  another  square,  overlapping  and  nearly 
coinciding  with  the  former.     Thus  every  point  of  the  object  is 
represented  by  its  square  on  the  screen ;  and  as  the  pencils  all 
cross  at  the  aperture,  the  image  formed  is  every  way  inverted.    It 
is  also  indistinct,  because  the  squares  overlap,  and  the  light  of 
contiguous  points  is  mingled  together.     If  the  orifice  is  smaller, 
the  image  is  less  luminous,  but  more  distinct,  because  the  pencils 
which  form  it  overlap  in  a  less  degree.     If  the  hole  is  circular,  or 
triangular,  or  of  irregular  form,  there  is  no  change  in  the  appear- 
ance of  the  image,  which  is  now  composed  of  small  circles,  or 
triangles,  or  irregular  figures,  whose  shape  is  completely  lost  by 
overlapping. 

455.  Effect  of  a  Convex  Lens  at  the  Aperture.— The 

image  will  become  distinct,  and  more  luminous  also,  if  the  aper- 
ture be  enlarged  to  a  diameter  of  two  or  three  inches,  and  then 
covered  by  a  convex  lens  of  the  proper  curvature.  The  image  will 
be  distinct,  because  the  rays  from  each  point  of  the  object  are  con- 
verged to  a  point  again,  and  luminous  in  proportion  as  the  lens 
has  a  larger  area  than  the  aperture  before  employed.  This  is  a 
real,  and  therefore  an  inverted  image  (Art.  385).  A  scioptic  ball 
is  a  sphere  containing  a  lens,  and  so  fitted  in  a  socket  that  it  can 
be  turned  in  any  direction,  and  thus  bring  into  the  room  the 


THE    EYE. 


2.95 


FIG.  280 


?'n"iges  of  different  parts  of  the  landscape.  The  camera  obscura  is  a 
darkened  room  furnished  with  a  scioptic  ball  and  adjustable  screen 
for  producing  distinct  pictures  of  external  objects. 

Instead  of  connecting  the  lens  with  the  wall  of  a  room,  it  is 
frequently  attached  to  a  portable  box  or  case,  within  which  the 
image  is  formed.  The  Daguerreotype,  or  photograph,  is  the  image 
produced  by  the  convex  lens,  and  rendered  permanent  by  the 
chemical  action  of  light  on  a  surface  properly  prepared.  The  lens 
for  photographic  purposes  needs  to  be  achromatic,  and  corrected, 
also,  as  far  as  possible,  for  spherical  aberration. 

456.  The  Eye. — The  eye  is  a  camera  obscura  in  miniature  ; 
we  find  here  the  darkened  room,  the  aperture,  the  convex  lens, 
and  the  screen,  with  inverted  images  of  exteraal  objects  projected 
on  it.     A  horizontal  section  of  the  eye  is  represented  in  Fig.  280. 

The  optical  apparatus  of  the  eye,  and  the  spherical  case  which 
incloses  it,  constitute 
what  is  called  the  eye- 
ball The  case  itself, 
except  about  a  sixth 
part  of  it  in  front,  is  a 
strong  white  substance, 
called,  on  account  of  its 
hardness,  the  sclerotic 
coat,  8,  S  (Fig.  280). 
In  the  front,  this  opaque 
coat  changes  to  a  per- 
fectly transparent  cover- 
ing, called  the  cornea, 
C,  C,  which  is  a  little 
more  convex  than  the 
sclerotic  coat.  The  in- 
creased convexity  of  the 

cornea  may  be  felt  by  laying  the  finger  gently  on  the  eyelid  when 
closed,  and  then  rolling  the  eye  one  way  and  the  other. 

The  bony  socket,  which  contains  the  eye,  is  of  pyramidal  form, 
its  vertex  being  some  distance  behind  the  eyeball ;  room  is  thus 
afforded  for  the  mechanism  which  gives  it  motion.  This  cavity, 
except  the  hemisphere  in  front  occupied  by  the  eye  itself,  is  tilled 
up  with  fatty  matter  and  with  the  six  muscles  by  which  the  eye- 
ball is  revolved  in  all  directions. 

457.  The  Interior  of  the  Eye.— Behind  the  cornea  is  a 
fluid,  A,  called  the  aqueous  humor.     In  the  back  part  of  this  fluid 
lies  the  iris,  I,  I,  an  opaque  membrane,  having  in  the  centre  of  it 
a  circular  aperture,  the  pupil,  through  which  the  light  enters. 


296  OPTICS. 

The  iris  is  the  colored  part  of  the  eye  ;  the  back  side  of  it  is  black. 
Directly  back  of  the  aqueous  humor  and  iris,  is  a  flexible  double 
convex  lens,  L,  called  the  crystalline  lens,  or  crystalline  humor, 
having  the  greatest  convexity  on  the  back  side.  The  large  space 
back  of  the  crystalline  is  occupied  by  the  vitreous  humor,  F,  a 
semi-liquid,  of  jelly-like  consistency.  Next  to  the  vitreous  humor 
succeed  those  inner  coatings  of  the  eye,  which  are  most  imme- 
diately concerned  in  vision.  First  in  order  is  the  retina,  R,  R,  on 
which  the  light  paints  the  inverted  pictures  of  external  objects. 
The  fibres  of  the  optic  nerve,  which  enter  the  ball  at  N,  are  spread 
all  over  the  retina,  and  convey  the  impressions  produced  there  to 
the  brain.  Outside  of  the  retina  is  the  choroid  coat,  c  li,  c  h, 
covered  with  a  black  pigment,  which  serves  to  absorb  all  the  light 
so  soon  as  it  has  passed  through  the  retina  and  left  its  impressions. 
The  choroid  is  inclosed  by  the  sclerotic  already  described.  The 
nerve-fibres,  which  are  spread  over  the  interior  of  the  retina,  are 
gathered  into  a  compact  bundle  about  one-tenth  of  an  inch  in 
diameter,  which  passes  out  through  the  three  coatings  at  the  back 
part  of  the  ball,  about  fifteen  degrees  from  the  axis,  XX,  on  the 
side  toward  the  other  eye.  M,  M  represent  two  of  the  muscles, 
where  they  are  attached  to  the  eyeball. 

458.  Vision. — The  index  of  refraction  for  the  cornea,  and  the 
aqueous  and  vitreous  humors,  is  just  about  the  same  as  that  for 
water ;  for  the  crystalline  lens,  the  index  is  a  little  greater.  The 
light,  therefore,  which  comes  from  without,  is  converged  principally 
on  entering  the  cornea,  and  this  convergency  is  a  little  increased 
both  on  entering  and  leaving  the  crystalline.  If  the  convergency 
is  just  sufficient  to  bring  the  rays  of  each  pencil  to  a  focus  on  the 
retina,  then  the  images  are  perfectly  formed,  and  there  is  distinct 
vision.  To  prevent  the  reflection  of  rays  back  and  forth  within 
the  chamber  of  the  eye,  its  walls  are  made  perfectly  black  through- 
out by  a  pigment  which  lines  the  choroid,  the  ciliary  processes,  and 
the  back  of  the  iris.  Telescopes  and  other  optical  instruments 
are  painted  black  in  the  interior  for  a  similar  purpose. 

The  cornea  is  prevented  from  producing  spherical  aberration 
by  the  form  of  a  prolate  spheroid  which  is  given  to  its  surface, 
and  the  crystalline,  by  a  gradual  increase  of  density  from  its  edge 
to  its  centre. 

459.  Adaptations.— By  the  prominence  of  the  cornea  rays 
of  considerable  obliquity  are  converged  into  the  pupil,  so  that  the 
eye,  without  being  turned,  has  a  range  of  vision  more  or  less  per- 
fect, through  an  angle  of  about  150°. 

The  quantity  of  light  admitted  into  the  eye  is  regulated  by  the 
size  of  the  pupil.  The  iris,  composed  of  a  system  of  circular  and 


ADAPTATIONS.  297 

radial  muscles,  expands  or  contracts  the  pupil  according  to  the 
intensity  of  the  light.  These  changes  are  involuntary  ;  a  person 
may  see  them  in  his  own  eyes  by  shading  them,  and  again  letting 
a  strong  light  fall  upon  them,  while  he  is  before  a  mirror. 

The  pupils  in  the  eyes  of  animals  have  different  forms  accord- 
ing to  their  habits  ;  in  the  eyes  of  those  which  graze,  the  pupil  is 
elongated  horizontally,  and  in  the  eyes  of  beasts  and  birds  of  prey, 
it  is  elongated  vertically. 

The  eyes  of  animals  are  adapted,  in  respect  to  their  refractive 
power,  to  the  medium  which  surrounds  them.  Animals  which 
inhabit  the  water  have  eyes  which  refract  much  more  than  those 
of  land  animals.  The  human  eye  being  fitted  for  seeing  in  air,  is 
unfit  for  distinct  vision  in  water,  since  its  refractive  power  is 
nearly  the  same  as  that  of  water,  and  therefore  a  pencil,  of  parallel 
rays  from  water  entering  the  eye  would  scarcely  be  converged  at 
all.  The  effect  is  the  same  as  if  the  cornea  were  deprived  of  all 
its  convexity. 

460.  Accommodation  to  Diminished  Distance.— It  has 

been  shown  (Art.  385),  that  as  an  object  approaches  a  lens,  its 
image  moves  away,  and  the  reverse.*  Therefore  in  the  eye  there 
must  be  some  change  in  order  to  prevent  this,  and  keep  the 
image  distinct  on  the  retina  while  the  object  varies  its  distance. 
In  a  state  of  rest,  the  eye  converges  to  the  retina  only  the  pencils 
of  parallel  rays,  that  is,  those  which  come  from  objects  at  great 
distances.  Eays  from  near  objects  diverge  so  much  that,  while  the 
eye  is  at  rest,  it  cannot  sufficiently  converge  them  so  that  they 
will  meet  on  the  retina ;  but  each  conical  pencil  is  cut  off  before 
reaching  its  focus,  and  all  the  points  of  the  object  are  represented 
by  overlapping  circles,  causing  an  indistinct  image.  The  change 
in  the  eye,  which  fits  it  for  seeing  near  objects  distinctly,  is  called 
accommodation.  This  is  effected  by  increasing  the  convexity  of 
the  crystalline  lens,  principally  the  front  surface.  The  ciliary 
muscle,  m,  m,  surrounds  the  crystalline,  and  is  attached  to  the 
sclerotic  coat  just  on  the  circle  where  it  changes  into  the  cornea. 
This  muscle  is  connected  with  the  edge  of  the  crystalline  by  the 
circular  ligament  which  surrounds  the  latter  and  holds  it  in  place. 
When  the  muscle  contracts,  it  relaxes  the  ligament,  and  the  crys- 
talline, by  its  own  elastic  force,  begins  to  assume  a  more  convex 
form,  as  represented  by  the  dotted  line.  The  eye  is  then  accom- 
modated for  the  vision  of  objects  more  or  less  near,  according  to 
the  degree  of  change  in  the  lens.  On  the  other  hand,  when  the 
ciliary  muscle  relaxes,  the  ligament  again  draws  upon  the  lens  to 
flatten  it,  and  adapt  it  for  the  view  of  distant  objects.  In  Fig. 
281  these  two  conditions  of  the  crystalline  are  more  distinctly 


OPTICS. 


FIG.  281. 


shown.  The  dotted  line  exhibits  the  shape  of  the  lens  when 
accommodated  for  seeing  near  objects.  Accompany- 
ing this  action  of  the  ciliary  muscle  is  that  of  the 
iris,  which  diminishes  the  pupil  for  near  objects,  so 
as  to  exclude  the  outer  and  more  divergent  rays. 
The  dotted  lines  in  front  of  the  iris  represent  its 
situation  when  pushed  forward  by  the  crystalline 
accommodated  for  near  objects. 

461.  Long-Sightedness. — As  life  advances, 
the  crystalline  becomes  harder  and  less  elastic.  It 
therefore  assumes  a  less  convex  form  when  the  liga- 
ment is  relaxed,  and  cannot  be  accommodated  to  so  short  dis- 
tances as  in  earlier  years  ;  and  at  length  it  remains  so  flattened  in 
shape  that  only  very  distant  objects  can  be  seen  distinctly.  The 
eye  is  then  said  to  be  long-sighted,  and  requires  a  convex  lens  to 
be  placed  before  it,  to  compensate  for  insufficient  convexity  in  the 
crystalline. 

There  are,  however,  cases  of  long-sightedness  in  early  life. 
Such  instances  are  found  to  be  the  result  of  an  oblate  form  of  the 
eyeball,  as  shown  in  Fig.* 
282;  it  is  too  short  from 
front  to  back  to  furnish 
room  for  the  convergency  of 
the  pencils,  and  they  are 
cut  off  by  the  retina  before 
reaching  their  focal  points. 
In  order  to  bring  the  dis- 
tinct image  forward  upon  the  retina,  convex  glasses  are  needed  in- 
such  cases,  just  as  for  the  eyes  of  most  people  when  advanced  in 
life.  As  the  term  long-sightedness  is  now  applied  to  this  abnor- 
mal condition  of  the  eye,  the  effect  of  age  upon  the  sight  is  more 
properly  called  old-sightedness. 


FIG.  282. 


FIG.  283. 


462.  Short-Sightedness.— The  eyes  of  the  short-sighted 

have  a  form  the  reverse  of 
that  just  described;  the  eye- 
ball is  elongated  from  cornea 
to  retina  (Fig.  283),  resem- 
bling a  prolate  spheroid,  so 
that  rays  parallel,  or  nearly 
so,  are  converged  to  a  point 
before  reaching  the  retina, 
and  after  crossing,  fall  on  it  in  a  circle  ;  and  the  image,  made  up- 
of  overlapping  circles  instead  of  points,  is  indistinct.  If  this- 


THE    BLIND    POINT.  .          299" 

elongation  of  the  eyeball  is  extreme,  an  object  must  be  brought 
very  near,  in  order  that  its  image  may  move  back  to  the  retina, 
and  distinct  vision  be  produced.  This  inconvenience  is  remedied 
by  the  use  of  concave  lenses,  which  increase  the  divergency  of  the 
rays  before  they  enter  the  eye,  and  thus  throw  their  focal  points 
further  back. 

In  the  normal  condition  of  the  eyes  in  early  life,  the  nearest 
limit  of  distinct  vision  is  about  five  inches.  This  limit  slowly 
increases  with  advance  of  life,  but  much  more  slowly  in  some 
cases  than  others,  till  it  is  at  an  indefinitely  great  distance.  The 
near  limit  of  distinct  vision  for  the  short-sighted  varies  from  fiv& 
down  to  two  inches,  according  to  the  degree  of  elongation  in  the 
eyeball. 

463.  Why  an  Object  is  Seen  Erect  and  Single.— The- 

image  on  the  retina  is  inverted;  and  that  is  the  very  reason  why 
the  object  is  seen  erect ;  the  image  is  not  the  thing  seen,  but  that 
by  means  of  which  we  see.  The  impression  produced  at  any  point 
on  the  retina  is  referred  outward  in  a  straight  line  through  a  point 
near  the  centre  of  the  lens,  to  something  external  as  its  cause ; 
and  therefore  that  is  judged  to  be  highest  without  us  which  make* 
its  image  lowest  on  the  retina,  and  the  reverse. 

An  object  appears  as  one,  though  we  see  it  by  means  of  two- 
images ;  but  this  is  only  one  of  many  instances  in  which  we  have 
learned  by  experience  to  refer  two  or  more  sensations  to  one  thing 
as  the  cause.  Provided  the  images  fall  on  parts  of  the  retina, 
which  in  our  ordinary  vision  correspond  with  each  other,  then  by 
experience  we  refer  both  impressions  to  one  object ;  but  if  we 
press  one  eye  aside,  the  image  falls  in  a  new  place  in  relation  to- 
the  other,  and  the  object  seems  double. 

464.  Indirect  Vision.— The  Blind  Point— To  obtain  a, 
clear  and  satisfactory  view  of  an  object,  the  axes  of  both  eyes  are 
turned  directly  upon  it,  in  which  case  each  image  is  at  the  centre 
of  the  retina.     But  when  the  light  from  an  object  is  exceedingly 
faint,  it  is  better  seen  by  indirect  vision,  that  is,  by  looking  to  a 
point  a  little  on  one  side,  and  especially  by  changing  the  direction 
of  the  eyes  from  moment  to  moment,  so  that  the  image  may  fall 
in  various  places  near  the  centre  of  the  retina.    Many  heavenly 
bodies  are  plainly  discerned  by  indirect  vision,  which  are  too  faint 
to  be  seen  by  direct  vision. 

In  the  description  of  the  eye  it  was  stated  that  the  retina,  as, 
well  as  the  choroid  and  the  sclerotic,  is  perforated  to  allow  the 
optic  nerve  to  pass  through.  At  that  place  there  is  no  vision,  and 
it  is  called  the  blind  point.  In  each  eye  it  is  situated  about  15° 
from  the  centre  of  the  retina  toward  the  other  eye.  Let  a  person 


300  OPTICS. 

close  his  right  eye,  and  with  the  left  look  at  a  small  but  conspicu 
ous  object,  and  then  slowly  turn  the  eye  away  from  it  toward  the 
right ;  presently  the  object  will  entirely  disappear,  and  as  he  looks 
still  further  to  the  right,  it  will  after  a  moment  reappear,  and  con- 
tinue in  sight  till  the  axis  of  the  eye  is  turned  70°  or  80°  from  it. 
The  same  experiment  may  be  tried  with  the  right  eye  in  the  oppo- 
site direction.  The  reason  why  people  do  not  generally  notice  the 
fact  till  it  is  pointed  out,  is  that  an  object  cannot  disappear  to  both 
eyes  at  once,  nor  to  either  eye  alone,  when  directed  to  the  object. 

465.  Continuance  of  Impressions. — The  impression  which 
a  visible  object  makes  upon. the  retina  continues  about  one-eighth 
or  one-ninth  of  a  second ;  so  that  if  the  object  is  removed  for 
that  length  of  time,  and  then  occupies  its  place  again,  the  vision 
is  uninterrupted.     A  coal  of  fire  whirled  round  a  centre  at  the 
rate  of  eight  or  nine  times  per  second,  appears  in  all  parts  of  the 
circumference  at  once.     When  riding  in  the  cars,  one  sometimes 
gets  a  faint  but  apparently  an  uninterrupted  view  of  the  land- 
scape beyond  a  board  fence,  by  means  of  successive  glimpses  seen 
through  the  cracks  between  the  upright  boards.     Two  pictures, 
on  opposite  sides  of  a  disk,  are  brought  into  view  together,  as 
parts  of  one  and  the  same  picture,- by  whirling  the  disk  rapidly 
on  one  of  its  diameters.     Such  an  instrument  is  called  a  thauma- 
irope.    The  pliantasmascope  is  constructed  on  the  same  principle. 
Several  pictures  are  painted  in  the  sectors  of  a  circular  disk, 
.representing  the  same  object  in  a  series  of  positions.     These  are 
viewed  in  a  mirror  through  holes  in  the  disk,  as  it  revolves  quickly 
in  its  own  plane.     Each  glimpse  which  is  caught  whenever  a  hole 
comes  before  the  eye,  presents  the  object  in  a  new  attitude  ;  and 
all  these  views  are  in  such  rapid  succession  that  they  appear  like 
one  object  going  through  the  series  of  movements. 

466.  Subjective   Colors. — There  are  impressions   on   the 
retina  of  another  kind,  which  are  produced  by  intense  lights  ;  they 
continue  longer,  and  are  in  respect  to  color  unlike  the  objects 
which  cause  them.     They  are  called  subjective  colors.    If  a  par- 
ticular part  of  the  retina  is  for  some  time  affected  by  the  image  of 
a  bright  colored  object,  and  then  the  eyes  are  shut,  or  turned 
upon  a  white  surface,  the  form  appears  to  remain,  but  the  color  is 
complementary  to  that  of  the  object;  and  its  continuance  is  for  a 
few  seconds  or  several  minutes,  according  to  the  vividness  of  the 
impression. 

That  portion  of  the  retina  upon  which  the  bright  colored 
image  was  formed  loses  sensitiveness  to  that  particular  color  after 
a  short  time,  and  when  white  light  falls  upon  the  retina  that  par- 
ticular spot  is  affected  by  the  complementary  color  only. 


DISTANCE    OF    BODIES.  301 

467.  Irradiation. — When  small  bodies  are  intensely  illumi- 
nated the  retina  is  affected  somewhat  beyond  their  proper  images 
upon  it,   and  the  bodies  consequently  appear  larger  than  they 
would  if  less  bright.     A  white  circle  upon  a  black  ground  looks 
larger  than  an  equal  black  circle  upon  a  white  ground.     At  new 
moon,  when  both  the  bright  and  the  dark  portions  are  visible,  the 
crescent  seems  to  be  a  part  of  a  larger  sphere  than  that  which  it 
accompanies. 

This  enlargement  of  the  image  is  called  irradiation. 

468.  Estimate  of  the  Distance  of  Bodies.— 

1.  If  objects  are  near,  we  judge  of  relative  distance  by  the  in- 
clination of  the  optic  axes  to  each  other.     The  greater  that  incli- 
nation is,  or,  which  is  the  same  thing,  the  greater  the  change  of 
direction  in  an  object,  as  it  is  viewed  by  one  eye  and  then  by  the 
other,  the  nearer  it  is.     If  objects  are  very  near,  we  can  with  one 
•eye  alone  judge  of  their  distance  by  the  degree  of  effort  required 
to  accommodate  the  eye  to  that  distance. 

2.  If  objects  are  known,  we  estimate   their  distance  by  the 
visual  angle  which  they  fill,  having  by  experience  learned  to 
associate  together  their  distance  and  their  apparent,  that  is,  their 
angular  size. 

3.  Our  judgment  of  distant  objects  is  influenced  by  their  clear- 
ness or  obscurity.     Mountains,  and  other  features  of  a  landscape, 
if  seen  for  the  first  time  when  the  air  is  remarkably  pure,  are 
•estimated  by  us  nearer  than  they  really  are ;  and  the  reverse,  if 
the  air  is  unusually  hazy. 

4.  Our  estimate  of  distance  is  more  correct  when  many  objects 
intervene.    Hence  it  is  that  we  are  able  to  place  that  part  of  the 
sky  which  is  near  the  horizon  further  from  us  than  that  which  is 
over  our  heads.     The  apparent  sky  is  not  a  hemisphere,  but  a 
flattened  semi-ellipsoid. 

469.  Magnitude  and  Distance  Associated.— Our  judg- 
ments of  distance  and  of  magnitude  are  closely  associated.    If 
objects  are  known,  we  estimate  their  distance  by  their  visual  angle, 
as  has  been  stated ;  but  if  unknown,  we  must  first  acquire  our 
notion  of  their  distance  by  some  other  means,  and  then  their 
visual  angle  gives  us  a  definite  impression  as  to  their  size.    And  if 
our  judgment  of  distance  is  erroneous,   a  corresponding  error 
attaches  to  our  estimate  of  their  magnitude.     An  insect  crawling 
slowly  on  the  window,  if  by  mistake  it  is  supposed  to  be  some 
rods  beyond  the  window,  will  appear  like  a  bird  flying  in  the  air, 
The  moon  near  the  horizon  seems  larger  than  above  us,  because 
we  are  able  to  locate  it  at  a  greater  distance. 


302  OPTICS. 

The  apparent  linear  dimensions  of  objects  are  directly  propor- 
tional to  their  actual  dimensions  and  inversely  proportional  to- 
their  distances  from  the  observer. 

470.  Binocular  Vision.— The  Stereoscope. — If  objects  are 
placed  quite  near  us,  we  obtain  simultaneously  two  views,  which 
are  essentially  different  from  each  other — one  with  one  eye,  and 
one  with  the  other.  By  the  right  eye  more  of  the  right  side,  and 
less  of  the  left  side,  is  seen,  than  by  the  left  eye.  Also,  objects  in 
the  foreground  fall  further  to  the  left  compared  with  distant  ob- 
jects, when  seen  with  the  right  eye  than  when  seen  with  the  left. 
And  we  associate  with  these  combined  views  the  form  and  extent 
of  a  body,  or  group  of  bodies,  particularly  in  respect  to  distance 
of  parts  from  us.  It  is,  then,  by  means  of  vision  with  two  eyes,  or 
binocular  vision,  that  we  are  enabled  to  get  accurate  perceptions, 
of  prominence  or  depression  of  surface,  reckoned  in  the  visual 
direction.  A  picture  offers  no  such  advantage,  since  all  its  parts 
are  on  one  surface,  at  a  common  distance  from  the  eyes.  But,  if 
two  perspective  views  of  an  object  should  be  prepared,  differing 
as  those  views  do,  which  are  seen  by  the  two  eyes,  and  if  the  right 
eye  could  then  see  only  the  right-hand  view,  and  the  left  eye  only 
the  left-hand  view,  and  if,  furthermore,  these  two  views  could  be 
made  to  appear  on  one  and  the  same  ground,  the  vision  would 
then  be  the  same  as  is  obtained  of  the  real  object  by  both  eyes. 
This  is  effected  by  the  stereoscope.  Two  photographic  views  are 
taken,  in  directions  which  make  a  small  angle  with  each  other, 
and  these  views  are  seen  at  once  by  the  two  eyes  respectively, 
through  a  pair  of  half-lenses,  placed  with  their  thin  edges  toward 
each  other,  so  as  to  turn  the  visual  pencils  away  from  each  other, 
as  though  they  emanated  from  one  object.  An  appearance  of 
relief  and  reality  is  thus  given  to  superficial  pictures,  precisely 
like  that  obtained  from  viewing  the  objects  themselves. 


CHAPTER    IX 

OPTICAL    INSTRUMENTS. 

471.  The  Camera  Lucida. — This  is  a  four-sided  prism,  so- 
contrived  as  to  form  an  apparent  image  at  a  surface  on  which  that 
image  may  be  copied,  the  surface  and  image  being  both  visible  at 
the  same  time.  It  has  the  form  represented  by  the  section  in. 


THE    MICROSCOPE. 


303 


FIG.  281 


Tig.  284;  A  =  90°,  0=  135°;  B  and  D,  of  any  convenient  size, 
their  sum  of  course  =  135°.  A  pencil  of  light  from  the  object 
M,  falling  perpendicularly  on  A  D,  proceeds  on,  and  makes,  with 
D  C,  an  angle  equal  to  the  complement  of  D.  After  suffer- 
ing total  reflection  at  G,  and  again  at 
H,  its  direction  H  E  is  perpendicular 
to  M  F.  For,  produce  M  F,  and  EH, 
till  they  intersect  in  /;  then,  since 
C  =  135°,  C  G  H+  C  H  G  =  45°  ;  but 
/  G  H=  2  C  G  H,  and  /  H  G  =  2 
CHG\  .:!  G  H+IffG  =  W°; 
«•.  /  =  90°.  Therefore  H  E  emerges 
at  right  angles  to  A  B,  and  is  not  re- 
fracted. Now,  if  the  pupil  of  the 
«ye  be  brought  over  the  edge  B,  so 
that,  while  E  H  enters,  there  may 
also  enter  a  pencil  from  the  surface 
at  M',  then  both  the  surface  M'  and 
the  object  M  will  be  seen  coinciding  with  each  other,  and  the  hand 
may  therefore  sketch  M  on  the  surface  at  M'.  The  reason  for 
two  reflections  of  the  light  is,  that  the  inversion  produced  by  one 
reflection  may  be  restored  by  the  second. 

One  of  the  most  useful  applications  of  the  camera  lucida  is  in 
connection  with  the  compound  microscope,  where  it  is  employed 
in  copying  with  exactness  the  forms  of  natural  objects,  too  small 
to  be  at  all  visible  to  the  naked  eye. 

472.  The  Microscope. — This  is  an  instrument  for  viewing 
minute  objects.     The  nearer  an  object  is  brought  to  the  eye,  the 
larger  is  the  angle  which  it  fills,  and  therefore  the  more  perfect  is 
the  view,  provided  the  rays  of  each  pencil  are  converged  to  a  point 
on  the  retina.     But  if  the  object  is  nearer  than  the  limit  of  dis- 
tinct vision,  the  eye  is  unable  to  produce  sufficient  convergency. 
If  the  letters  of  a  book  are  brought  close  to  the  eye,  they  become 
blurred  and  wholly  illegible.    But  let  a  pin-hole  be  pricked  through 
paper,  and  interposed  between  the  eye  and  the  letters,  and,  though 
faint,  they  are  distinct  and  much  enlarged.    The  distinctness  is 
owing  to  the  fact  that  the  outer  rays,  which  are  most  divergent, 
.are  excluded,  and  the  eye  is  able  to  converge  the  few  central  rays 
of  each  pencil  to  a  focus.     The  letters  appear  magnified,  because 
they  are  so  near,  and  fill  a  large  angle.     The  microscope  utilizes 
these  excluded  rays,  and  renders  the  image  not  only  large  and  dis- 
tinct, but  luminous. 

473.  The  Single  Microscope. — The  single  microscope  is 
merely  a  convex  lens.     It  aids  the  eye  in  converging  the  rays, 


304:  OPTICS. 

•which  come  from  an  object  placed  between  the  lens  and  its  focus, 
so  that  a  distinct  and  luminous  image  may  be  formed  on  the 
retina. 

Let  P  Q  (Fig.  285)  be  the  object,  and  p  q  its  image.  The 
image  is  at  the  distance  of  distinct  vision  (24  cm.),  and  subtends  a 

larger  angle   than   the   object 
FIG.  285.  would   at   the  same  distance  ; 

it    therefore    appears    larger. 

Since  the  eye  is  always  placed 

on   the    lens,    the   magnifying 

p  q         C   q      T    . 
power    is  ^L  =  __L     Let 

F  =  focal  length  of  the  lens, 
G    Q  =  p,  and    C  q  =  —  q 
(negative    because    the    image    is  virtual).      Art.   383    gives  us 

1       i_i 
F       p       q 

whence  JL=J_  +  l  =  ^_|  =  |-i  =  magnifying   power. 

Since  q  =  24  cm.,  the  magnifying  power  is  greater  the  smaller 
the  focal  length  of  the  lens.  E.g.,  if  F  =  3  cm.  the  magnifica- 
tion is  9. 

Though  the  focal  distance  of  a  lens  may  be  made  as  small  as 
we  please,  yet  a  practical  limit  to  the  magnifying  power  is  very 
soon  reached. 

1.  'She  field  of  view,  that  is,  the  extent  of  surface  which  can  be 
seen  at  once,  diminishes  as  the  power  is  increased. 

2.  Spherical  aberration  increases  rapidly,  because  the  outer 
rays  are  very  divergent.     Hence  the  necessity  of  diminishing  the 
aperture   of  the   lens,   in   order   to   exclude   the  most  divergent 
rays. 

3.  it  is  more  difficult  to  illuminate  the  object  as  the  focal 
length  of  the  lens  becomes  less  ;   and  this  difficulty  becomes  a 
greater  evil  on  account  of  the  necessity  of  diminishing  the  aperture 
in  order  to  reduce  the  spherical  aberration. 

To  lessen  the  spherical  aberration,  two  or  more  plano-convex 
lenses  are  used,  close  together.  When  this  is  done,  the  plane 
face  of  each  lens  is  turned  towards  the  object.  Although  each 
lens  is  less  convex  than  the  simple  lens  which  together  they  re- 
place, yet  their  joint  magnifying  power  is  as  great,  and  with  a  less 
amount  of  spherical  aberration,  since  the  first  lens  draws  towards 
the  axis  the  rays  which  fall  on  the  second  lens.  This  combination 
of  lenses  is  known  as  Wollasloris  doublet. 

Magnify  ing-glasses  are  single  microscopes  of  low  power,  such  -tis 


MODERN    IMPROVEMENTS. 


305 


are  used  by  watchmakers.    Lenses  of  still  lower  power  and  several 
inches  in  diameter  are  used  for  viewing  pictures. 

474.  The  Compound  Microscope. — It  is  so  called  because 
it  consists  of  two  parts,  an  object-glass,  by  which  a  real  and  mag- 
nified image  is  formed,  and  an  eye-glass,  by  which  that  image  is 
again  magnified.    Its  general  principle  may  be  explained  by  Fig. 
286,  in  which  a  b  is  the  small  object,  c  d  the  object- 
glass,  and  ef  the  eye-glass.     Let  a  b  be  a  little 

beyond  the  principal  focus  of  c  d,  and  then  the 

image  g  h  will  be  real,  on  the  opposite  side  of  c  d, 

and  larger  than  a  b.    Now  apply  ef  as  a  single 

microscope  for  viewing  g  Ji,  as  though  it  were  an 

object  of  comparatively  large  size.     Let  g  h  be  at 

the  principal  focus  of  ef,  so  that  the  rays  of  each 

pencil  shall  be  parallel ;  they  will,  therefore,  come 

t0  the  eye  at  k,  from  an  apparent  image  on  the 

same  side  as  the  real  one,  g  h ;  and  the  extreme 

pencils,    e  k,  f  k,  if   produced   backward,    will 

include  the  image  between  them,  e  k  f  being  the  angle  which 

it  fills. 

475.  The  Magnifying  Power. — The  magnifying  power  of 
the  compound   microscope   is    estimated  by  compounding  two 
ratios  ;  first,  the  distance  of  the  image  from  the  object-glass,  to 
the  distance  of  the  object  from  the  same  ;  and  secondly,  the  limit 
of  distinct  vision  to  the  distance  of  the  image  from  the  eye-glass. 
For  the  image  itself  is  enlarged  in  the  first  ratio  (Art.  385) ;  and 
the  eye-glass  enlarges  that  image  in  the  second  ratio  (Art.  473). 
The  advantage  of  this  form  over  the  single  microscope  is  not  so 
much  that  a  great  magnifying  power  is  obtained,  as  that  a  given 
magnifying  power  is  accompanied  by  a  larger  field  of  view. 

476.  Modern  Improvements. — Great  improvements  have 
been  made  in  the  compound  microscope,  principally  by  combining 
lenses  in  such  a  manner  as  greatly  to  reduce  the  chromatic  and 
spherical  aberrations.     The  object-glass  generally  consists  of  one, 
two,  or  three  achromatic  pairs  of  lenses.     The  eye-piece  usually 
contains  two  plano-convex  lenses,  a  combination  which  is  found 
to  be  the  most  favorable  for  diminishing  the  spherical  aberration, 
and  for  enlarging  the  field  of  view. 

In  Fig.  287  let  a  b  be  the  object,  C  an  achromatic  lens,  called 
the  objective,  D  the  field  lens  (so  called  because  it  enlarges  the 
field  of  view  by  bending  the  pencil  which  would  come  to  a  focus 
at  «',  and  pass  below  the  eye  lens,  so  that  it  may  come  to  a  focus 
at  a",  and  thence  pass  into  the  eye  lens),  E  the  eye  lens  which 


306 


OPTICS. 


renders  the  rays  of  the  pencil  so  nearly  parallel  that  the  eye  re- 
ceives them  as  though  coming  from  the  point  a'". 


FIG.  287. 


FIG.  288. 


The  method  of  determining  the  magnifying  power  given  in 
the  last  paragraph,  is  not  applicable  in  this  form  of  instrument, 
but  an  experimental  determination  is  made  as  follows  :  A  very 
finely-divided  scale  is  placed  under  the  microscope ;  a  mirror,  from 
which  a  small  part  of  the  silvering  has  been  removed,  is  placed 
near  the  eye-piece  at  an  angle  of  45°  with  the  axis  of  the  instru- 
ment, and  behind  this  at  the  distance  of  ordinary  distinct  vision, 
about  ten  inches,  and  visible  through  the  unsilvered  part  of  the 
mirror,  is  placed  a  second  scale  like  the  first ;  the  number  of 
divisions  of  the  second  scale  covered  by  one  division  seen  through 
the  microscope  by  reflection  gives  the  power.  Any  change  in  the 

relative  positions  of  the  lenses, 
changes  the  magnifying  power. 

4T7.  The  Projecting  Lan- 
tern.— Projecting  lanterns,  ste- 
reopticons,  and  magic  lanterns 
are  all  constructed  upon  the  same 
plan.  The  principle  involved  is 
shown  in  Fig.  288.  An  electric 
arc-lamp,  lime  or  other  powerful 
light  is  placed  in  a  box.  This 
box  is  provided  with  holes  which 
allow  ventilation  but  prevent  the 
escape  of  light.  In  one  side  of 
the  box,  at  the  same  height  as 
the  arc,  is  an  opening  in  which  is 
placed  a  condenser,  L.  The  con- 
denser consists  of  two  plano- 
convex lenses  with  their  convex  surfaces  opposed  to  each  other. 
(A  double  convex  lens  is  sometimes  used  in  place  of  the  combina- 


THE     TELESCOPE. 


307 


lion.)  These  lenses  are  of  short  focal  length  and  serve  to  con- 
centrate the  rays  from  the  lamp  upon  the  slide  P.  Upon  the  slide 
is  photographed  the  picture  or  object  to  be  projected.  After  pass- 
ing the  slide,  the  rays  are  brought  to  a  focus  on  a  distant  screen 
by  the  projecting  lens  L'.  This  lens  is  so  adjusted  that  the  slide  P 
and  the  screen,  upon  which  it  is  projected,  are  at  conjugate  foci. 
This  is,  therefore,  a  little  outside  the  focal  length  of  L'.  It  can 
be  readily  seen  that  the  diameter  of  the  image  on  the  screen  will 
bear  the  same  ratio  to  the  diameter  of  the  slide  as  their  respective 
distances  from  the  lens  L'. 


478.  The   Telescope.—  The  telescope   aids  in  viewing  dis- 
tant bodies.     An  image  of  the  distant  body  is  first  formed  in  the 
principal  focus  of  a  convex  lens  or  a  concave  mirror  ;  and  then  a 
microscope  is  employed  to  magnify  that  image  as  though  it  were  a 
small  body.     The  image  is  much  more  luminous  than  that  formed 
in  the  eye,  when  looking  at  the  heavenly  body,  because  there  is 
concentrated  in  the  former  the  large  beam  of  light  which  falls 
upon  the  lens  or  mirror,  while  the  latter  is  formed  by  the  slender 
pencil  only  which  enters  the  pupil  of  the  eye.     If  the  image  in  a 
telescope  is  formed  by  a  lens,  the  instrument  is  called  a  refracting 
telescope  ;  but  if  by  a  mirror,  a  reflecting  telescope. 

479.  The   Astronomical   Telescope.  —  This  is  the  most 
simple  of  the  refracting  telescopes,  consisting  of  a  lens  to  form  an 
image  of  the  heavenly  body,  and  a  single  microscope  for  magnify- 
ing that  image.     The  former  is  called  the  object-glass,  the  latter 
the  eye-glass. 

Let  a  (Fig.  289)  be  the  image  of  some  point  of  a  heavenly 
body,  the  divergent  rays  from  which,  marked  A  A'  A,  are  practi- 


FIG. 


cally  parallel,  and  b  the  image  of  the  point  B  B'  B.  As  the  rays 
forming  these  images  are  parnllel  rays,  a  b  is  at  the  principal 
focus  of  the  object-glass  0.  The  eye  lens  E  receives  the  diver- 


308  OPTICS. 

gent  pencils  from  a  and  J,  bends  them  so  that  they  enter  the  eye 
as  parallel  or  nearly  parallel  beams  coming  apparently  from  the 
direction  of  a'  and  V .  The  image  a  b  is  situated  at  the  principal 
focus  of  E,  the  distance  between  the  lenses  0  and  E  being  the 
sum  of  their  principal  focal  distances. 

480.  The  Powers  of  the  Telescope. — The  magnifying 
power  of  the  astronomical  telescope  is  expressed  by  the  ratio  of  the 
focal  distance  of  the  object-glass  to  that  of  the  eye-glass.  For  (Fig. 
289)  the  object  as  it  would  be  seen  by  the  eye  if  placed  at  0  fills 
the  angle  A'  0  B'  between  the  axes  of  its  extreme  pencils.  But, 
since  the  axes  cross  each  other  in  straight  lines  at  the  optic  centre 
of  the  lens,  A'  0  B'  =  a  0  b.  Therefore,  to  an  eye  placed  at  the 
object-glass,  the  image,  a  b,  appears  just  as  large  as  the  object ; 
while  at  the  eye-glass  it  appears  as  much  larger  in  diameter  as  the 
distance  is  less. 

Since  no  simple  eye  lenses  are  used,  and  as  the  equivalent 
power  of  the  compound  eye-piece  is  not  readily  found,  the  follow- 
ing practical  method  of  finding  the  power  of  an  astronomical 
telescope  is  of  use  : 

Adjust  the  eye-piece  so  that  a  sharp  and  clear  view  of  some 
very  distant  object  may  be  obtained  ;  remove  the  object-glass,  and 
in  its  place  put  an  opaque  card  disk  in  which  is  cut  an  opening  of 
the  shape  of  a  very  flat  isosceles  triangle,  whose  base  is  nearly  as 
long  as  the  diameter  of  the  object-glass  ;  receive  an  image  of  this 
opening  upon  a  translucent  glass  or  paper  screen  held  close  to  the 
eye-piece,  and  measure  the  base  of  the  image  very  exactly ;  the 
length  of  the  base  of  the  opening  divided  by  the  base  of  its  image 
is  the  power.  If  any  of  the  rays  from  the  opening  are  cut  off  by 
diaphragms  in  the  tube  the  imperfection  of  the  image  will  make 
known  the  difficulty,  which  must  be  removed. 

The  field  of  view  is  determined  thus  :  Direct  the  telescope  to  a 
star  on  or  near  the  celestial  equator,  and  note  the  time  in  seconds 
which  the  star  occupies  in  passing  across  the  diameter  of  the  field 
of  view ;  divide  this  time  by  4  and  the  quotient  will  be  the  diame- 
ter of  the  field  in  minutes  of  arc,  because  a  star  on  the  equator 
moves  through  one  minute  of  arc  in  four  seconds  of  time. 

The  ilium inatiny  power  is  important  for  objects  which  shed  a 
very  feeble  light  on  account  of  their  immense  distance.  This 
power  depends  on  the  size  of  the  beam,  that  is,  on  the  aperture  of 
the  object-glass. 

The  defining  power  is  the  power  of  giving  a  clear  and  sharply- 
defined  image,  without  which  both  the  other  powers  are  useless. 
And  it  is  the  power  of  producing  a  well-defined  image  which 
limits  both  of  the  other  powers.  For  every  attempt  to  increase 


THE     TERRESTRIAL     TELESCOPE.  30!) 

the  magnifying  power  by  giving  a  large  ratio  to  the  focal  lengths 
of  the  object-glass  and  the  eye-glass,  or  to  increase  the  illuminat- 
ing power  by  enlarging  the  object-glass,  increases  the  difficulties 
in  the  way  of  getting  a  perfect  image.  These  difficulties  are  three 
— the  spherical  aberration  (Art.  388),  the  chromatic  aberration 
(Art.  402),  and  unequal  densities  in  the  glass.  The  third  diffi- 
culty is  a  very  serious  one,  especially  in  large  lenses. 

481.  The  Terrestrial  Telescope. — In  order  to  secure  sim- 
plicity, and  thus  the  highest  excellence,  in  the  astronomical  tele- 
scope, the  image  is  allowed  to  be  inverted,  which  circumstance  is 
of  no  importance  in  viewing  heavenly  bodies.  But,  for  terrestrial 
objects,  it  would  be  a  serious  inconvenience ;  and,  therefore,  a  ter- 
restrial telescope,  or  spy-glass,  has  additional  lenses  for  the  purpose 
of  forming  a  second  image,  inverted,  compared  with  the  first,  and, 
therefore,  erect,  compared  with  the  object.  In  Fig.  290,  m  m  m 
represent  a  pencil  of  rays  from  the  top  of  a  distant  object,  and 

FIG.  290. 


n  n  n  from  the  bottom  ;  A  B,  the  object-glass  ;  m  n,  the  first 
image;  CD,  the  first  eye-glass,  which  converges  the  pencils  of 
parallel  rays  to  L.  Instead  of  placing  the  eye  at  L,  the  pencils 
are  allowed  to  cross  and  fall  on  the  second  eye-glass,  E  F,  by  which 
the  rays  of  each  pencil  are  converged  to  a  point  in  the  second 
image,  m'  n',  which  is  viewed  by  the  third  eye-glass,  G  H.  The 
second  and  third  lenses  are  commonly  of  equal  focal  length,  and 
add  nothing  to  the  magnifying  power. 

Such  instruments  are  usually  of  a  portable  size,  and  hence  the 
aberrations  are  corrected  with  comparative  ease,  by  the  methods 
already  described.  The  spy-glass,  for  convenient  transportation, 
is  made  of  a  series  of  tubes,  which  slide  together  in  a  very  com- 
pact form. 

If  the  lenses  CD  and  E  Fare  of  the  same  power  they  do  not 
affect  the  power  of  the  telescope,  which  may  then  be  represented 

F 


as  in  the  astronomical  telescope  by 


—?. 


To  determine  the  power  practically,  look  at  some  distant  scale 
of  equal  parts,  a  brick  wall  for  instance,  and  keeping  both  eyes 


310  OPTICS. 

open,  note  how  many  bricks  as  seen  by  the  unaided  eye  are  covered 
by  the  image  of  one  brick  as  seen  through  the  telescope  ;  this 
number  so  covered  is  the  expression  for  the  power.  If  one  space, 
for  instance,  seen  through  the  telescope,  covers  twenty  spaces  seen 
with  the  unaided  eye,  the  telescope  magnifies  twenty  diameters. 

482.  Galileo's  Telescope.— This  was  the  first  form  of 
telescope,  having  been  invented  by  Galileo,  whose  name  it  there- 
fore bears.  It  differs  from  the  common  astronomical  telescope  in 
having  for  the  eye-glass  a  concave  instead  of  a  convex  lens,  which 
receives  the  rays  at  such  a  distance  from  the  focus  to  which  they 
tend,  as  to  render  them  parallel. 

Thus  the  rays  m  m  m  (Fig.  291),  from  a  point  at  the  top  of 
the  object,  are  converged  by  the  object-glass  0  towards  a  focus  a; 

FIG.  291. 


but  before  meeting  at  a  they  fall  upon  a  concave  eye-lens  E,  and 
are  rendered  parallel  or  slightly  divergent,  as  though  they  came 
from  a  point  in  the  direction  indicated  by  M.  The  point  from 
which  the  rays  m  m  m  proceeded,  and  its  virtual  image  M,  are 
both  on  the  same  side  of  the  axis  of  the  instrument,  and  there  is 
no  inversion. 

It  is  obvious  that,  since  the  pencils  diverge,  only  the  central 
ones,  within  the  size  of  the  pupil,  can  enter  the  eye.  This  cir- 
cumstance exceedingly  limits  the  field  of  view,  and  unfits  the 
instrument  for  telescopic  use.  It  is  employed  for  opera-glasses, 
having  a  power  usually  of  only  two  or  three  diameters. 

F 

The  expression  for  the  power  is  -^-,  as  in  the  preceding  forms. 

The  power  may  be  very  readily  determined  practically  as  in  the 
case  of  the  terrestrial  telescope. 

483.  The  Gregorian  Telescope. — This  is  the  most  fre- 
quent form  of  reflecting  telescope,  and  receives  its  name  from  the 
inventor,  Dr.  Gregory,  of  Scotland. 

Let  A  (Fig  292)  be  a  point  of  a  very  distant  body  from  which 
rays,  practically  parallel,  fall  upon  the  large  concave  mirror  M, 
which  is  perforated  through  the  middle  o  o' ;  these  are  converged 
to  the  principal  focus  a,  and  passing  this  point,  diverging  again, 
are  received  by  the  small  concave  reflector  R  of  short  focus,  and 


THE    NEWTONIAN     TELESCOPE. 


311 


are  made  to  converge  to  a,  forming  a  real  image  ;  thence  the  rays 
diverge  once  more,  and  falling  upon  the  eye-glass  E,  are  refracted 
as  though  they  came  from  an  object  in  the  direction  A'. 

The   Cassegrainian    telescope   differs  from  the  Gregorian  in 
having  a  convex  reflector  in  place  of  the  concave  R ;  this  is  so 

FIG.  292. 


placed  as  to  receive  the  rays  before  they  reach  the  focus  «,  and, 
by  rendering  them  less  convergent,  bring  them  to  a  focus  at  the 
place  of  the  image  a',  but  upon  the  same  side  of  the  axis  as  «. 

484.  The  Newtonian  Telescope.— R  is  a  concave  reflec- 
tor (Fig.  293),  F &  plane  mirror  called  a,  fat,  E  the  convex  eye- 
glass. Eays  from  some  point  A  of  a  distant  object  are  converged 
by  R  towards  the  principal  focus  a  ;  they  are  intercepted  by  the 
flat  jFand  turned  aside,  without  change  of  convergency,  to  the 

FIG.  293. 


focus  «',  passing  which  they  fall  upon  the  eye-lens  E,  and  enter 
the  eye  as  though  they  came  from  the  direction  A'.  The  magni- 
fying power  = 

focal  length  of  reflector 

focal  length  of  eye-glass ' 

485.  The  Herschelian  Telescope. — Sir  William  Herschel 
modified  the  Newtonian  by  dispensing  with  the  small  reflector  F, 
and  inclining  the  large  speculum  R,  so  as  to  form  the  image  near 
the  edge  of  the  tube,  where  the  eye-glass  is  attached.  Thus,  the 


312 


OPTICS. 


observer  is  situated  with  his  back  to  the  object.  The  speculum  of 
HerscheFs  telescope  was  about  four  feet  in  diameter,  and  weighed 
more  than  2,000  pounds,  and  its  focal  length  was  forty  feet.  The 
Earl  of  Rosse  has  since  constructed  a  Herschelian  telescope  having 
an  aperture  of  six  feet,  and  a  focal  length  of  fifty  feet.  The  mag- 
nifying power  is  the  same  as  in  the  Newtonian. 


486.  Eye-pieces,  or  Oculars. — The  negative,  or  Huyghe- 
nian,  eye-piece  consists  of  two  plano-convex  lenses  of  crown  glass, 
F  and  E  (Fig.  294),  the  convex  surfaces  being  turned  toward  the 
object-glass.  A  pencil  of  rays  from  the  object-glass,  converging 

to  a  principal  focus  a,  is  bent 

FIG.  294.  from  its  course  by  F  and 

brought  to  a  focus  a,  half- 
way between  the  two  lenses. 
The  image  formed  at  a'  is 
then  viewed  by  the  eye-lens 
E  as  usual. 

This  eye-piece  is  called  negative  because  it  is  adapted  to  rays 
already  converging.  The  focal  length  of  F  is  three  times  that  of 
E,  and  the  distance  between  the  lenses  is  one-half  the  sum  of  the 
focal  lengths. 

This  combination  is  also  achromatic.  In  practice  the  object- 
glass,  used  in  connection  with  the  Huyghens'  eye-piece,  is  an 
achromatic  lens,  which,  however,  is  over-corrected.  The  rays  after 
passing  it  are  refracted  by  F  so  as  to  form  a  series  of  colored 
images  around  a.  Owing  to  the  over-correction  the  red  image  is 
formed  at  the  left  of  a  and  the  violet  at  the  right.  These  images 
are  then  made  to  coalesce  by  the  refi-action  of  the  lens  E.  Huy- 
ghens was  not  aware  of  this  peculiarity  of  his  eye-piece. 

The  positive,  or  Ramsden,  eye-piece  consists  of  two  plano- 
convex lenses  E'  and  E  (Fig.  295)  with  the  convex  surfaces 
turned  toward  each  other.  The  rays  from  the  object-glass  are 
focussed  at  a  and  thence 

pass  to  the   eye  as   indi-  FIG.  295. 

cated  in  the  figure.  Two 
lenses  are  used  instead 
of  one,  in  order  more 
easily  and  perfectly  to 

correct  spherical  aberration.  E'  and  E  are  of  equal  focal  lengths, 
and  the  distance  between  them  is  two-thirds  the  focal  length  of 
one  of  them.  This  combination  is  not  achromatic.  It  is  always 
used  when  spicier  lines  are  placed  in  the  focus  of  the  object-glass 
for  purposes  of  exact  measurement. 


PART    VI. 


CHAPTER  I. 

EXPANSION  BY  HEAT.— THE    THERMOMETER. 

487.  Nature  of  Heat. — There  is  abundant  reason  for  be- 
lieving that  heat  consists  of  exceedingly  minute  and  rapid  vibra- 
tions of  ordinary  matter  and  of  the  ether  which  fills  all  space.     It 
is  to  be  regarded  as  one  of  the  modes  of  motion,  which  may  be 
caused  by  any  kind  of  force,  and  which  may  be  made  a  measure 
of  that  force.     Heat  affects  only  one  of  our  senses,  that  of  feeling. 
Its  increase  produces  the  sensation  of  warmth,  and  its  diminution 
that  of  cold. 

488.  Expansion  and  Contraction  by  Heat  and  Cold. 

— It  is  found  to  be  a  fact  almost  without  exception,  that  as  bodies 
are  heated  they  are  expanded,  and  that  they  contract  as  they  are 
cooled.  It  is  easy  to  conceive  that  the  vibratory  motion  of  the 
several  molecules  of  a  body  compels  them  to  recede  from  each 
other,  and  to  recede  the  more  as  the  vibration  becomes  more 
violent.  Although  the  change  in  magnitude  is  generally  very 
small,  yet  it  is  rendered  visible  by  special  contrivances,  and  is 
made  the  means  of  measuring  temperature. 

489.  Expansion    of  Solids.— When  the  expansion  of  a 
solid  is  considered  simply  in  one  dimension,  it  is  called  linear 
expansion ;  in  two  dimensions  only,  superficial  expansion  ;  in  all 
three  dimensions,  cubical  expansion. 

The  linear  expansion  of  a  metallic  rod  is  readily  made  visible 
by  an  instrument  called  the  pyrometer,  which  magnifies  the  mo- 
tion. The  end  A  of  the  rod  A  B  (Fig.  296)  is  held  in  place  by 
a  screw.  The  end  B  rests  against  the  short  arm  of  the  lever  C,  the 
longer  arm  of  which  bears  on  the  arm  D  of  the  long  bent  lever 
D  E ;  this  serves  as  an  index  to  the  graduated  arc  E  F.  The  long 
metallic  dish  G  (f,  being  raised  on  the  hinges  H  H,  so  as  to 


314:  HEAT. 

enclose  the  bar  A  B,  and  then  filled  with  hot  water,  the  bar 
instantly  expands,  and  raises  the  index  along  the  arc  E  F. 

FIG.  296. 


490.  Coefficient  of  Expansion. — The  coefficient  of  linear 
expansion  of  a  given  substance  is  the  fractional  increase  of  its 
length,  when  its  temperature  is  raised  one  degree.  But  since  this 
increase  is  generally  somewhat  greater  at  higher  temperatures,  the 
coefficients  of  expansion  given  in  tables  usually  refer  to  a  tempera- 
ture at  or  near  the  freezing  point  of  water.  Thus  the  coefficient 
of  expansion  for  silver  is  0.000019097  ;  by  which  is  meant  that  a 
silver  bar  one  foot  long  at  0°  C.  becomes  1.000019097  ft.  in  length 
at  1°  C. 

The  coefficient  of  superficial  expansion  is  twice,  and  that  of 
cubical  expansion  three  times  as  great  as  the  coefficient  of  linear 
expansion.  For,  suppose  c  to  be  the  coefficient  of  linear  expan- 
sion ;  then  if  the  edge  of  a  cube  is  1,  and  the  temperature  is  raised 
1°,  the  edge  becomes  1  +  c,  and  the  area  of  one  side  becomes 
(1  +  cY  =  I  +  2c  4-  c*,and  the  volume  (1  +  c)3  =  l  +  3c  +3c*+c*. 
But  as  c  is  very  small,  the  higher  powers  may  be  neglected,  and 
the  area  is  l  +  2c,  and  the  volume  is  1  +  3c  ;  that  is,  the  coefficient 
of  superficial  expansion  is  2c,  and  that  of  cubical  expansion  is  3c, 
as  stated  above. 


491.  The  Coefficient  of  Expansion  differs  in  different 
Substances. — Copper  expands  nearly  twice  as  much  as  platinum 
for  a  given  increase  of  temperature ;  the  ratio  of  expansion  in 
steel  and  brass  is  about  as  61  to  100.  This  ratio  is  employed  in 
the  construction  of  the  compensation  pendulum  (Art.  164). 

If  two  thin  slips  of  metal  of  different  expansibility  be  soldered 
together  so  as  to  make  a  slip  of  double  thickness,  it  will  bend  one 
way  and  the  other  by  changes  of  temperature.  If  it  is  straight 
at  a  certain  temperature,  heating  will  bend  it  so  as  to  bring  the 
most  expansible  metal  on  the  convex  side  ;  and  cooling  will  bend 
it  in  the  opposite  direction ;  and  the  degree  of  flexure  will  be  ac- 
cording to  the  degree  of  change  in  temperature.  Compensation 
in  clocks  and  watches  is  sometimes  effected  on  this  plan.  If  the 


THE    THERMAL    FORCE.  315. 

compound  slip  has  the  form  of  a  helix,  with  the  most  expansible 
metal  on  the  inside,  heating  will  begin  to  uncoil  it,  and  cooling, 
to  coil  it  closer.  A  very  sensitive  thermometer,  known  as  Bre- 
guet's  thermometer,  is  constructed  on  this  principle. 

As  notable  exceptions  to  the  general  rule  that  solids  expand 
when  heated,  may  be  mentioned  stretched  India-rubber,  and  also 
Eose's  fusible  metal,  an  alloy  of  2  parts  bismuth,  1  part  lead,  and 
1  part  tin  ;  the  latter  compound  expands  up  to  44°  C.,  then 
rapidly  contracts  up  to  69°  C.,  which  is  the  temperature  of  maxi- 
mum density,  and  again  expands  till  it  melts  at  94°  C. 

492.  The  Strength  of  the  Thermal  Force.— It  is  found 
that  the  force  exerted  by  a  body,  when  expanding  by  heat  or  con- 
tracting by  cold,  is  equal  to  the  mechanical  force  necessary  to- 
expand  or  compress  the  body  to  the  same  degree.     The  force  is 
therefore  very  great.     If  the  rails  were  to  be  fitted  tightly  end  to- 
end  on  a  railroad,  they  would  be  forced  out  of  their  places  by 
expansion  in  warm  weather,  and  the  track  ruined.     The  tire  of  a 
carriage  wheel  is  heated  till  it  is  too  large,  and  then  put  upon  the 
wheel ;  when  cool,  it  draws  together  the  several  parts  with  great 
firmness.     In  repeated  instances,  the  walls  of  a  building,  when 
they  have  begun  to  spread  by  the  lateral  pressure  of  an  arched 
roof,  have  been  drawn  togetner  by  the  force  of  contraction  in  cool- 
ing.   A  series  of  iron  rods  being  passed  across  the  building  through, 
the  upper  part  of  the  walls,  and  broad  nuts  being  screwed  upon 
the  ends,  the  alternate  bars  are  expanded  by  the  heat  of  lamps, 
and  the  nuts  tightened.     Then,  when  they  cool,  they  draw  the 
walls  toward  each  other.     The  remaining  bars  are  then  treated  in 
the  same  manner,  and  the  process  is  repeated  till  the  walls  are 
restored  to  their  vertical  position  and  secured. 

493.  Expansion  of  Liquids. — As  liquids  have  no  perma- 
nent form,   the  coefficient    of    expansion    for    them  is  always 
understood  to  be  that  of  cubical  expansion.     There  is  a  practical 
difficulty  in  the  way  of  finding  the  coefficient  for  liquids,  because 
they  must  be  enclosed  in  some  solid,  which  also  expands  by  heat. 
Hence,  the  apparent  expansion  must  be  corrected  by  allowing  for 
the  expansion  of  the  inclosing  solid,  before  the  coefficient  of 
absolute  expansion  is  known. 

This  fact  is  illustrated  by  the  following  experiment.  Fill  the 
bulb  and  part  of  the  stem  of  a  large  thermometer  tube  with  a 
colored  liquid,  and  then  plunge  the  bulb  quickly  into  hot  water ; 
the  first  effect  is,  that  the  liquid  falls,  as  if  it  were  cooled  ;  after  a 
moment  it  begins  to  rise,  and  continues  to  do  so  till  it  attains  the 
temperature  of  the  hot  water.  The  first  movement  is  caused  by 
the  expansion  of  the  glass,  which  is  heated  so  as  to  enlarge  its 


316  HEAT. 

capacity  and  let  down  the  liquid  before  the  heat  has  penetrated 
the  latter.  It  is  obvious  that  what  is  rendered  visible  in  this  case, 
must  always  be  true  when  a  liquid  is  heated — namely,  that  the 
vessel  itself  is  enlarged,  and  therefore  that  the  rise  of  the  liquid 
shows  only  the  difference  of  the  two  expansions.  Ingenious 
methods  have  been  devised  for  obtaining  the  coefficients  of  ab- 
solute expansion  of  liquids,  and  the  results  are  to  be  found  in 
tables  on  this  subject. 

From  the  examination  of  such  tables  we  learn  :  (1)  That 
liquids  expand  more  than  solids  for  a  given  increase  of  tem- 
perature ;  (2)  that  the  coefficient  of  expansion  increases  with  the 
rise  of  temperature ;  (3)  that  the  more  volatile  the  liquid  the  more 
rapidly  will  it  expand  for  a  given  rise  of  temperature.  • 

494.  Exceptional  Case. — There  is  a  very  important  excep- 
tion to  the  general  law  of  expansion  by  heat  and  contraction  by 
cold,  in  the  case  of  water  just  above  the  freezing  point.     If  water 
be  cooled  down  from  its  boiling  point,  it  continually  contracts  till 
it  reaches  39.1°  F.  or  3.94°  C.,  when  it  begins  to  expand,  and 
continues  to  expand  till  it  freezes  at  32°  F.  or  0°  C.     On  the 
other  hand,  if  water  at  32°  F.  be  heated,  it  contracts  till  it  reaches 
39.1°  F.  or  3.94°  C.,  when  it  commences  to  expand.     Therefore 
the  density  of  water  is  greatest  at  the  point  where  this  change 
occurs.     Different  experimenters  vary  a  little  as  to  its  exact  place, 
but  it  is  usually  called  4°  C.,  or  39°  F. 

The  importance  of  this  exception  is  seen  in  the  fact  that  ice 
forms  on  the  surface  of  water,  and  continues  to  float  until  it  is 
again  melted.  As  the  cold  of  winter  comes  on,  the  upper 
stratum  of  a  lake  grows  more  dense  and  sinks  ;  and  this  process 
continues  till  the  temperature  of  the  surface  reaches  39°  F.,  when 
it  is  arrested.  Below  that  temperature  the  surface  grows  lighter 
as  it  becomes  colder,  till  ice  is  formed,  which  shields  the  water 
beneath  from  the  severe  cold  of  the  air  above. 

As  in  solids  so  in  liquids,  the  thermal  force  is  very  great. 
Suppose  mercury  to  be  expanded  by  raising  its  temperature  one 
degree,  it  would  require  more  than  300  pounds  to  the  square  inch 
to  compress  it  to  its  former  volume. 

495.  Expansion  of  Gases. — The  gases  expand  by  heat 
more  rapidly  and  more  regularly  than  solids  and  liquids.     The 
large  expansion  and  contraction  of  air  is  made  visible  by  immers- 
ing the  open  end  of  a  large  thermometer  tube  in  colored  liquid. 
When  the  bulb  is  warmed,  bubbles  of  air  are  forced  out  and  rise  to 
the  top  of  the  liquid  ;  when  it  is  cooled,  the  air  contracts  and  the 
liquid  rises  rapidly  in  the  tube. 

Gases,  at  a  constant  pressure,  expand  much  more  than  liquids 


THE    THERMOMETER.  317 

or  solids  for  a  given  increment  of  temperature.  All  gases,  at 
temperatures  much  above  that  of  liquefaction,  have  almost  exactly 
the  same  coefficients  of  expansion.  The  coefficient  of  expansion 
for  air  is  ^^  from  0°  C.  to  1°  C.  This  coefficient  increases  slightly 
Avith  increase  of  temperature  and  pressure. 

To  find  the  volume  of  any  gas  at  0°  C.,  let  v  be  the  known 
volume  at  t°  C.,  also  let  v'  be  the  required  Volume  at  0°  C.,  then 
v  =  v'  (1  +  ^  0> 

from  which  we  have     v'  —  z r— 

1  +  UTS  f 

If  there  is  a  change  of  pressure,  then,  since  the  tensions  or 
pressures  are  inversely  as  the  volumes,  the  temperatures  being  the 
same  (Art.  237),  we  have 

v"  :  v'  ::  p'  :  p, 

in  which  v"  is  the  volume  at  0°  C.  and  barometric  pressure  of 
760  mm.,  v'  the  volume  at  pressure  p,  and  p  the  normal  pressure 
760  mm. ;  from  which  we  get,  by  substituting  for  v'  its  value  above, 


1  +  irhr  t      P 

496.  The  Thermometer. — This  instrument  measures  the 
degree  of  heat,  or  the  temperature,  of  the  medium  around  it,  by 
the  expansion  and  contraction  of  some  substance.  The  substance 
commonly  employed  is  mercury.  The  liquid,  being  inclosed  in  a 
glass  bulb,  can  expand  only  by  rising  in  the  fine  bore  of  the 
stem,  where  very  small  changes  of  volume  are  rendered  visible. 
A  scale  is  attached  to  the  stem  for  reading  the  degrees  of  tem- 
perature. 

The  graduation  of  the  thermometer  must  begin  with  the  fixing 
of  two  important  points  by  natural  phenomena,  the  melting  of  ice 
and  boiling  of  water.  When  the  bulb  is  plunged  into  powdered 
ice,  the  point  at  which  the  column  settles  is  the  freezing-point 
of  the  thermometer.  And  if  it  is  placed  in  steam  under  the 
mean  atmospheric  pressure,  the  mercury  indicates  the  boiling- 
'  point.  Between  these  two  points,  namely  0°  and  100°  C.,  there 
must  be  100°,  and  the  scale  is  graduated  accordingly.  As  the 
bore  of  the  tube  is  not  likely  to  be  exactly  equal  in  all  parts, 
the  length  of  the  degrees  should  vary  inversely  as  the  area  of  the 
<;ross-section.  The  necessary  correction  is  determined  by  moving 
a  short  column  of  mercury  along  the  different  parts  and  compar- 
ing the  lengths  occupied  by  it.  The  degrees  in  the  several  parts 
must  vary  in  the  ratio  of  these  lengths. 

The  zero  of  the  scale  tends  to  rise  for  some  time  after  the 
thermometer  is  made,  the  change  amounting  to  more  than  2°  in 


318  HEAT. 

some  instances,  and  therefore  the  instrument  should  not  be  used 
for  at  least  six  months  after  construction.  The  zero  may  also  be 
displaced  by  subjecting  the  instrument  to  high  temperatures. 

497.  Different  Systems   of  Graduation.— There  are  in 
use  three  kinds  of  thermometer  scale,  Fahrenheit's,  Reaumur's, 
and  the  Centigrade  or  Celsius.      In  Fahrenheit's,  the  freezing 
point  of  water  is  called  32°,  and  the  boiling  point,  212° ;  in  Reau- 
mur's, the  freezing  point  is  called  0°,  and  the  boiling  point  80°; 
in  the  Centigrade,  the  freezing  point  0°,  and  the  boiling  point 
100°.     In  a  scientific  point  of  view,  the  Centigrade  is  preferable 
to  either  of  the  others,  but  Fahrenheit's  is  generally  used  in  this 
country.     The  letter  F.,  R,  or  C.,  appended  to  a  number  of  de- 
grees, indicates  the  scale  intended.     In  this  country,  F.  is  under- 
stood if  no  letter  is  used. 

498.  To   Reduce  from  one  Scale  to  Another. — Since 
the  zero  of  Fahrenheit's  scale  is  32°  below  the  freezing  point,  while 
in  both  of  the  others  it  is  at  the  freezing  point,  32°  must  always 
be  subtracted  from  any  temperature  according  to  Fahrenheit,  in 
order  to  find  its  relation  to  the  zero  of  the  other  scales.     Then, 
since  212°  -  32°  (=  180°)  F.  are  equal  to  80°  R,  and  to  100°  C., 
the  formula  for  changing  F.  to  R  is  $  (F.  —  32)  =  R  ;  and  for 
changing  F.  to  C.,  it  is  £  (F.  —  32)  =  C.    Hence,  to  change  .R 
to  F.,  we  have  f  R  +  32  =  F. ;   and  to  change  C  to  F.,  f  C. 
-}-  32  =  F. 

Mercury  congeals  at  about  — 38.8°  C.;  therefore,  for  tempera- 
tures lower  than  that,  alcohol  is  used,  which  does  not  congeal 
at  any  known  temperature. 

Above  100°  C.  the  indications  of  the  mercurial  thermometer 
are  not  exact. 

499.  Absolute  Zero  of  Temperature.— At  a  tempera- 
ture of  273°  C.  the  volume  of  a  gas  is  double  its  volume  at  0°  C. 
(Art.  495).     Suppose  that  instead  of  raising  the  temperature,  we 
lower  it ;  for  a  fall  from  0°  to  —1°  C.  the  volume  contracts  ^-j, 
and  for  a  fall  of  273°  it  must  contract  f|f ;  that  is  to  say,  the 
volume  would  disappear  entirely.     That  the  contraction  would  go 
on  to  — 273°  C.  is  not  asserted  ;  but  on  the  supposition  that  the  law 
of  contraction  would  hold,  we  fix  the  temperature  — 273°  as  that  at 
which  all  vibrations  would  cease,  and  at  which  consequently  there 
could  be  no  heat  whatever.     The  absolute  zero  more  exactly  given 
is  —273.7°  C.,  and  —460.66°  F.  The  absolute  temperature  is  found 
by  adding  these  readings,  with  signs  changed,  to  the  respective 
readings  of  the  mercurial  thermometer.    As  both  Fahrenheit  and 


CONDUCTION    OF    HEAT    BY    SOLIDS.  319 

Centigrade  thermometers  are  in  use,  both  will  be  referred  to  in 
the  text,  as  indicated,  that  the  student  may  become  familar  with 
both  systems  of  graduation. 


\J 


CHAPTER    II. 

PASSAGE    OF    HEAT    THROUGH    MATTER    AND    SPACE. 

600.  Heat  is  Communicated  in  Several  Ways. — 

1.  By  conduction.    This  is  the  slow  progress  of  the  vibratory 
motion  from  places  of  higher  to  places  of  lower  temperature  in 
the  same  body. 

2.  By  convection.    This  mode  of  communication  takes  place 
only  in  fluids.    When  the  particles  are  expanded  by  heat,  they  are 
pressed  upward  by  others  which  are  colder  and  therefore  specifi- 
cally heavier.     Heat  is  thus  conveyed  from  place  to  place  by  the 
motion  of  the  heated  matter,  though  the  ultimate  transfer  of  heat 
may  still  take  place  by  conduction. 

3.  By  radiation.     Heat  is  said  to  be  radiated  when  the  vibra- 
tory motion  is  transmitted  from  the  source  with  great  swiftness 
through  the  ether  which  fills  space.    Its  velocity  is  the  same  as 
that  of  light.    The  motion  is  propagated  in  straight  lines  in  every 
direction,  and  each  line«is  called  a  ray  of  heat.     We  feel  the  rays 
of  heat  from  the  sun  or  a  fire,  when  no  object  intervenes  between 
it  and  ourselves. 

501.  Conduction  of  Heat  by  Solids.— Conducted  heat 
passes  through  bodies  very  slowly,  and  yet  at  very  different  rates 
in  different  bodies.  Those  in  which  heat  is  conducted  most 
rapidly,  are  called  good  conductors,  as  the  common  metals  ;  those 
in  which  it  passes  slowly,  are  called  poor  conductors,  as  glass  and 
wood.  In  general,  the  bodies  which  are  good  conductors  of  heat, 
are  also  good  conductors  of  electricity  ;  thus  calling  the  conduc- 
tivity for  electricity  E,  and  for  heat  H,  and  using  silver  as  the 
standard,  we  find — 


Silver E  =  100  H  =  100 

Copper "      90  "      90 

<}old "      59  "      53 

Brass...              .     "      22  "       24 


Iron E  =  15  H  =  16 

Lead "      9  "      8 

German  silver "      8  "      8 

Bismuth  . .             .    "      1  "1 


Let  rods  of  different  metals  and  other  substances,  A,  B,  C, 


320 


HEAT. 


FIG.  297. 


&c.  (Tig.  297),  all  of  the  same  length,  be  inserted  with  water- 
tight joints  in  the  side  of  a  wooden  vessel.  Then  attach  by  wax  a 
marble  under  the  end  of  each  rod,  and  fill  the  vessel  with  boiling 

water.  The  marbles  will  fall 
by  the  melting  of  the  wax,  not 
at  the  same,  but  at  different 
times,  showing  that  the  heat 
reaches  some  of  them  sooner 
than  others.  It  will  be  seen, 
however,  in  the  chapter  on 
specific  heat,  that  the  order  in 
which  they  fall  is  not  neces- 
sarily the  order  of  conducting 
power. 

The  amount  of  heat  conducted  through  a  thin  lamina  is 
directly  proportional  to  the  area,  to  the  time  during  which  it 
flows,  to  the  difference  of  temperature  at  the  two  surfaces,  and  to 
the  conductivity  of  the  substance ;  and  is  inversely  proportional 
to  the  thickness  of  the  lamina. 

502.  Effects  of  Molecular  Arrangement. — Organic  sub- 
stances usually  conduct  heat  poorly ;  and  bodies  having  a  struc- 
tural arrangement  which  differs  in  different  directions,  are  not 
likely  to  conduct  equally  well  in  all  directions.  Thus,  let  two 
thin  plates  be  cut  from  the  same  crystal,  one,  A  (Fig.  298),  per- 


6 


FIG.  298. 


pendicular.  and  the  other,  B,  parallel  to  the  optic  axis.  Let  a 
hole  be  drilled  through  the  centre  of  each,  and  after  a  lamina  of 
wax  has  been  spread  over  the  crystal,  let  a  hot  wire  be  inserted  in 
it.  On  the  plate  A,  the  melting  of  the  wax  will  advance  in  a 
circle,  showing  equal  conducting  power  in  all  directions  in  the 
transverse  section.  In  the  plate  B,  it  will  advance  in  an  elliptical 
form,  the  major  axis  being  parallel  to  the  optic  axis  of  the  crystal, 
proving  the  best  conduction  to  be  in  that  direction. 

A  block  of  wood  cut  from  one  side  of  the  trunk  of  a  tree,  con- 
ducts most  perfectly  in  the  direction  of  the  fibre,  and  least  in  a 
direction  which  is  tangent  to  the  annual  rings  and  perpendicular 
to  the  fibre,  and  in  an  intermediate  degree  in  the  direction  of  the 
radius  of  the  rings. 


DIFFERENCE    IN    CONDUCTIVE    POWER.       321 

503.  Conduction  by  Fluids. — Fluids,  both  liquid  and  gas- 
eous, are  in  general  very  poor  conductors.     Water,  for  example, 
can  be  made  to  boil  at  the  top  of  a  vessel,  while  a  cake  of  ice  is 
fastened  within  it  a  few  inches  below  the  surface.     If  thermom- 
eters are  placed  at  different  depths,  while  the  water  boils  at  the 
top,  there  is  discovered  to  be  a  very  slight  conduction  of  heat  down- 
ward.     The  gases  conduct  even  more  imperfectly  than  liquids. 

It  will  be  seen  hereafter  (Art.  505)  that  a  mass  of  fluid  becomes 
heated  by  convection,  not  by  conduction. 

504.  Illustrations  of  Difference  in  Conductive  Power. 

— In  a  room  where  all  articles  are  of  equal  temperature,  some  feel 
much  colder  than  others,  simply  because  they  conduct  the  heat 
from  the  hand  more  rapidly;  painted  wood  feels  colder  than 
woolen  cloth,  and  marble  colder  still.  If  the  temperature  were 
higher  than  that  of  the  blood,  then  the  marble  would  seem  the 
hottest,  and  the  cloth  the  coolest,  because  of  the  same  difference 
of  conduction  to  the  hand. 

Our  clothing  does  not  impart  warmth  to  us,  but,  by  its  non- 
conducting property,  prevents  the  vital  warmth  from  being  wasted 
by  radiation  or  conduction.  If  the  air  were  hotter  than  our  blood, 
the  same  clothing  would  serve  to  keep  us  cool. 

A  pitcher  of  water  can  be  kept  cool  much  longer  in  a  hot  day, 
if  wrapped  in  a  few  thicknesses  of  cloth ;  for  these  prevent  the 
heat  of  the  air  from  being  conducted  to  the  water.  In  the  same 
way  ice  may  be  prevented  from  melting  rapidly. 

The  vibrations  of  heat,  like  those  of  sound,  are  greatly  inter- 
rupted in  their  progress  by  want  of  continuity  in  the  material. 
Any  substance  is  rendered  a  much  poorer  conductor  by  being  in 
the  condition  of  a  powder  or  fibre.  Ashes,  sand,  sawdust,  wool, 
fur,  hair,  &c.,  owe  much  of  their  non-conducting  quality  to  the 
innumerable  surfaces  which  heat  must  meet  with  in  being  trans- 
mitted through  them. 

Davy's  safety  lamp  is  a  practical  application  of  conduction. 
A  wire  gauze  surrounds  the  lamp,  and  the  air  which  supplies  the 
flame  with  oxygen  can  only  reach  it  by  passing  through  the  gauze. 
A  naked  flame  would  ignite  the  fire  damp  of  the  mines ;  but 
though  the  fire  damp  may  ignite  after  passing  the  gauze  and  may 
fill  the  whole  lamp  with  a  body  of  flame,  yet,  owing  to  the  cooling 
effected  by  the  conduction  of  the  wires,  the  gases  on  the  outside 
are  not  raised  to  the  temperature  of  ignition  ;  thus  warning,  and 
time  for  escape  from  danger,  are  given. 

505.  Convection  of  Heat. — Liquids  and  gases  are  heated 
almost  entirely  by  convection.     As  heat  is  applied  to  the  sides 
and  bottom  of  a  vessel  of  water,   the  heated  particles  become 


322 


HEAT. 


FIG.  299. 


specifically  lighter,  and  are  crowded  up  by  heavier  ones  which 
take  their  place.  There  is  thus  a  constant  circulation  going  on 
which  tends  to  equalize  the  temperature  of 
the  whole,  by  bringing  the  hot  portions 
into  contact  with  the  colder,  and  thus 
greatly  facilitating  the  conduction  of  heat 
among  the  molecules. 

This  motion  is  made  visible  in  a  glass 
vessel,  by  putting  into  the  water  some 
opaque  powder  of  nearly  the  same  density 
as  water.  Ascending  currents  are  seen  over 
the  part  most  heated,  and  Descending  cur- 
rents in  the  parts  farthest  from  the  heat,  as 
represented  in  Fig.  299.  The  ocean  has  per- 
petual currents  caused  in  a  similar  manner. 
The  hottest  portions  flow  away  from  the 
tropical  toward  the  polar  latitudes,  while  at 
greater  depth  the  cold  waters  of  high  lati- 
tudes flow  back  toward  the  tropics. 

For  a  like  reason,  the  air  is  constantly  in  motion.  The  atmos- 
pheric currents  on  the  earth  have  been  considered  in  Chapter  III. 
•of  Pneumatics. 

506.  Determination  of  the  Temperature   of  Water 
at  its  Maximum  Density. — The  apparatus  used  by  Joule  in 
his  research  is  represented  in  outline  in  Fig.  300,  in  which  A  and 
B  are,  cylinders  4£  feet  high  and  6  inches  in 
•diameter  ;  the  open  trough  C  connects  them  at 
top,  and  a  large  tube,  with  stop-cock  D,  con- 
nects them  at  the  bottom.     When  the  cylinders 
were  filled  so  that  there  was  a  free  flow  through 
the  trough  C,  any  difference  of  density  in  A 
and  B  would  produce  a  convection   current 
through  D  and  C,  and  the  existence  of  such 
current  in  C  was  made  known  by  the  motion  of 
a  small  glass  bulb,  of  nearly  the  specific  gravity 
of  water,  floating  in  the  trough.     A  very  slight 
difference   of    density   between   the   water  in 
A  and   B,   gave   motion    to  the    bulb  in   C. 
The   cock  D  being  closed,   the    temperatures 
of  A  and  B  were  adjusted  so  that  one  should  be  above  and  the 
other  below  that  of  maximum  density.      Having  recorded  the 
'temperatures,  D  was  opened,  and  any  difference  of  density  would 
be  shown  by  a  motion  of  the  bulb  towards  the  denser  column. 
By  carefully  adjusting  the  temperatures,  so  that  upon  opening  D 


RADIATION     OF     HEAT.  323 

no  motion  in  the  trough  C  should  result,  a  pair  of  temperatures 
was  obtained,  corresponding  to -the  same  density.  From  a  series 
of  such  pairs,  the  differences  of  which  were  made  successively 
smaller,  Joule  fixed  the  temperature  of  maximum  density  at 
39.1°  F.  or  3.94°  C.,  very  nearly. 

507.  Radiation  of  Heat.— Radiation  of  heat  is  the  com- 
munication of  the  vibrations  of  the  heated  body  to  the  ether  sur- 
rounding it,  by  which  the  waves  of  heat  are  transmitted  in 
the  manner  already  explained  in  the  article  Light.  Heat  rays 
differ  from  rays  of  light  only  in  wave  length,  and  are  capable  of 
reflection,  refraction,  interference,  and  polarization.  A  body  not 
hot  enough  to  send  forth  rays  affecting  the  optic  nerve,  still  sends 
out  heat  rays,  nor  can  any  body  be  so  cold  as  not  to  radiate  heat 
at  all. 

The  intensity  of  heat  radiated  from  a  given  source,  is  governed 
by  the  three  following  laws  : 

1.  The  intensify  of  radiated  heat  varies  as  the  temperature  of 
the  source. 

2.  It  varies  inversely  as  the  square  of  the  distance. 

3.  It  grows  less,  while  the  inclination  of  the  rays  to  the  surface 
of  the  radiant  grows  less. 

The  truth  of  these  laws  is  ascertained  by  a  series  of  careful 
experiments.  But  the  second  may  be  proved  mathematically 
from  the  fact  of  propagation  in  straight  lines,  as  in  sound  and 
light.  For  the  heat,  as  it  advances  in  every  direction  from  the 
radiant,  is  spread  over  spherical  surfaces  which  increase  as  the 
squares  of  the  distances  ;  therefore  the  intensities  must  grow  less 
in  the  same  ratio ;  that  is,  the  intensities  vary  inversely  as  the 
squares  of  the  distances. 

The  radiating  porver  of  a  given  body  depends  on  the  condition 
of  its  surface. 

If  a  cubical  vessel  filled  with  hot  water  have  one  of  its  vertical 
sides  coated  with  lamp  black,  another  with  mica,  a  third  with 
tarnished  lead,  and  the  fourth  with  polished  silver,  and  the 
heat  radiated  from  these  several  sides  be  concentrated  upon  a 
thermometer  bulb,  the  ratio  of  radiation  will  be  found  nearly  as 
follows  : 
Lamp  black 100 


Mica 80 


Tarnished  lead 45 

Polished  silvej- 12 


Polished  metals  generally  radiate  feebly  ;  and  this  explains  the 
familiar  fact  that  hot  liquids  retain  their  temperature  much  better 
in  bright  metallic  vessels  than  in  dark  or  tarnished  ones. 

When  the  temperature  of  a  body  is  gradually  raised,  not  only 


324  HEAT. 

are  new  kinds  of  radiations  produced,  whose  wave  lengths  are 
smaller  than  those  already  emitted,  but  the  intensity  of  existing 
radiations  also  increases.  A  white-hot  body  emits  more  red  rays 
than  a  red-hot  body,  and  more  non-luminous  rays  than  a  non- 
luminous  body. 

508.  Equalization  of  Temperature. — Eadiation  is  going 
on  continually  from  all  bodies,  more  rapidly  in  general  from  those 
most  heated  ;  and  therefore  there  is  a  constant  tendency  toward 
an  equal  temperature  in  all  bodies.     A  system  of  exchange  goes 
on,  by  which  the  hotter  bodies  grow  cool,  and  the  colder  ones 
grow  warm,  till  the  temperature  of  all  is  the  same.     But  this 
equality  does  not  check  the  radiation ;  it  still  goes  forward,  each 
body  imparting  to  every  other  as  much  heat  as  it  receives  from  it, 
the  radiations  emitted  and  absorbed  by  either  body  being  equal 
not  only  in  total  heating  effect,  but  being  the  same  in  the  inten- 
sity, wave  length,  and  plane  of  polarization  of  every  component 
part  of  either  radiation. 

509.  Reflection  of  Heat.— When  rays  of  heat  meet  the 
surface  of  a  body,  some  of  them  are  reflected,  passing  off  at  the 
same  angle  with  the  perpendicular  on  the  opposite  side.     But 
others  pass  into  the  body,  and  are  said  to  be  absorbed  by  it.     It  is 
true  of  waves  of  heat  as  of  all  other  kinds  of  vibration,  that  when 
they  meet  a  new  surface  and  are  reflected,  the  angle  of  incidence 
equals  the  angle  of  reflection,  and  that  their  intensity  after  reflec- 
tion is  weakened. 

If  a  person,  when  near  a  fire,  holds  a  sheet  of  bright  tin  so  as 
to  see  the  light  of  the  fire  reflected  by  it,  he  will  plainly  perceive 
that  heat  is  reflected  also.  And  if  any  sound  is  produced  by  the 
fire,  as  the  crackling  of  combustion,  or  the  hissing  of  steam  from 
wood,  the  reflection  of  the  sound  is  likewise  heard.  This  simple 
experiment  proves  that  waves  of  sound,  of  heat,  and  of  light,  all 
follow  the  same  law  of  reflection. 

510.  Heat  Concentrated  by  Reflection.— Let  two  pol- 
ished reflectors,  M  and  ^(Fig.  301),  having  the  form  of  concave 
paraboloids,  be  placed  ten  or  fifteen  feet  apart,  with  their  axes  in 
the  same  straight  line,  and  let  a  red-hot  iron  ball  be  in  the  focus 
A  of  one,  and  an  inflammable  substance,  as  phosphorus,  in  the 
focus  B  of  the  other ;  then  the  latter  will  be  set  on  fire  by  the 
heat  of  the  ball.     The  rays  diverging  from  A  to  M  are  reflected 
in  parallel  lines  to  N,  and  then  converged  to  B. 

If,  instead  of  phosphorus,  the  bulb  of  a  thermometer  is  put  in 
the  focus  B,  a  high  temperature  is  of  course  indicated  on  the  scale. 
Now  remove  the  hot  ball  from  A,  and  put  in  its  place  a  lump  of 


ABSORPTION     OF    HEAT.  325 

Ice ;  then  the  thermometer  at  B  sinks  far  below  the  temperature 

of  the  room.     This  last  experiment  does  not  prove  that  cold  is 

reflected  as  well  as  heat,  but  confirms  what  was  stated  (Art.  508)P 

FIG.  301. 


that  all  objects  radiate  to  one  another  till  their  temperatures  are 
equalized.  The  ice  radiates  only  a  little  heat,  which  is  reflected 
to  the  thermometer,  but  the  latter  radiates  much  more,  which  is 
reflected  to  the  ice,  so  that  the  temperature  of  the  thermometer 
rapidly  sinks. 

511.  Absorption  of  Heat. — The  radiant  heat  which  falls  on  a 
body  and  is  not  reflected  or  transmitted,  is  absorbed.     The  absorb- 
ing power  in  a  body  is  found  to  be  in  general  equal  to  its  radiating 
power.     It  is  very  noticeable  that  bodies  equally  exposed  to  the 
radiant  heat  of  the  sun  or  a  fire,  become  very  unequally  heated. 
A  white  cloth  on  the  snow,  under  the  sunshine,  remains  at  the 
surface ;  a  black  cloth  sinks,  because  it  absorbs  heat,  and  melts 
the  snow  beneath  it.     Polished  brass  before  a  fire  remains  cold  ; 
dark,  unpolished  iron,  is  soon  hot. 

Lamp  black  reflects  little  of  the  radiation  which  falls  on  it ; 
nearly  the  whole  is  absorbed. 

Polished  silver  reflects  the  greater  part  of  the  radiations  falling 
upon  it,  absorbs  only  about  2J  per  cent.,  and  transmits  none. 

Eock  salt  reflects  less  than  8  per  cent,  of  the  radiation  it 
receives,  absorbs  almost  none,  and  transmits  92  per  cent. 

512.  Diathermancy. — Substances  which  transmit  heat  rays, 
without  themselves  becoming  hot,  are  called  Diathermanous ; 


326  HEAT. 

those  which  are  heated  by  the  transmission  are  said  to  be  Atlier- 
manous. 

Kadiant  heat  passes  freely  through  the  atmosphere  as  well  as 
through  vacant  space.  The  air  is  therefore  said  to  be  diathermal; 
it  is  also  transparent,  since  it  permits  light  to  pass  freely  through 
it.  But  there  are  substances  which  allow  the  free  transmission  of 
the  waves  of  light,  but  not  those  of  heat ;  and  there  are  others 
through  which  waves  of  heat  can  freely  pass,  but  not  those  of 
light. 

Water  and  glass,  which  are  almost  perfectly  transparent  to 
the  faintest  light,  will  not  transmit  the  vibrations  of  heat  unless 
they  are  very  intense.  If  an  open  lamp-flame  shines  upon  a  thin 
film  of  ice,  while  nearly  the  whole  of  the  light  is  transmitted,  only 
6  per  cent,  of  the  heat  can  pass  through. 

A  plate  of  rock  salt,  one-tenth  of  an  inch  thick,  will,  as  shown 
in  the  last  paragraph,  transmit  92  per  cent,  of  the  heat  of  a 
lamp;  and  if  it  be  coated  with  lampblack  so  thick  as  to  stop 
light  completely,  the  heat  is  still  transmitted  with  almost  no 
diminution. 

Prisms  and  lenses  of  rock  salt  have  been  used  in  illustrating 
refraction  of  heat,  just  as  glass  prisms  and  lenses  are  used  in  the 
case  of  luminous  rays. 


/  CHAPTER    III. 

SPECIFIC  HEAT.— CHANGES  OF  CONDITION.— LATENT  HEAT. 

513.  Specific  Heat. — The  energy  of  a  single  molecule  of  any- 
chemical  element,  at  a  given  temperature,  is  the  same  as  for  a 
molecule  of  any  other  element  at  the  same  temperature.  To  raise 
the  temperature  requires  an  addition  of  energy  from  an  external 
source.  Molecules  of  different  substances  have  different  weights. 
Hence  a  gram  of  light  molecules  would  have  more  energy  at  a 
given  temperature  than  a  gram  of  heavier  ones  at  the  same  tem- 
perature, for  there  would  be  a  greater  number  of  the  lighter  mole- 
cules. The  amount  of  heat  energy  which  must  be  supplied  to  one 
gram  of  a  substance,  that  its  temperature  may  be  raised  from  0° 
to  1°  C.,  is  termed  the  specific  heat  of  that  substance.  , 

The  quantity  of  heat  or  heat  energy  is  measured  in  gram 
calories.  A  calorie  is  the  heat  necessary  to  raise  the  temperature 
of  a  gram  of  water  from  0°  to  1°  C. 


SPECIFIC  ,HEAT. 


327 


The  specific  heats  of  nearly  all  solids  and  liquids  increase  with 
temperature. 

The  mean  specific  heats  of  a  few  substances  between  0°  and 
100°  C.  are  given  below  to  show  how  greatly  they  differ. 


Water 1.000 

Glass 0.190 

Iron 0.113 

Nickel 0.110 

Copper .-. 0.094 


Silver 0.057 

Tin 0.056 

Mercury 0.034 

Gold 0.032 

Lead 0.032 


The  specific  heat  of  water  is  greater  than  that  of  any  other  known 
substance,  and  therefore  it  is  made  the  standard  of  comparison. 
The  great  specific  heat  of  water  moderates  the  changes  of  tem- 
perature upon  islands  and  upon  the  sea-coast. 

If  a  body  expands  by  rise  of  temperature,  part  of  the  heat 
energy  supplied  to  it  is  used  in  performing  internal  and  external 
work — the  internal  work  of  separating  the  molecules  and  the  ex- 
ternal work  against  atmospheric  pressure.  The  specific  heat  of  a 
gas,  which  is  confined  to  a  constant  volume  during  a  rise  of  tem- 
perature (and  must  thus  suffer  an  increase  of  pressure),  is  less 
than  if  allowed  to  expand,  i.e.,  at  constant  pressure.  With  constant 
volume  no  external  or  internal  work  is  performed,  and  thus  less 
heat  is  required  to  raise  the  temperature.  These  two  specific  heats 
for  gases  bear  the  ratio  of  1.41  :  1.  This  ratio  is  the  constant 
inserted  in  the  formula  for  the  velocity  of  sound  (Art.  284). 


514.  Method  of  Finding  Specific  Heat. — The  following  is 
one  of  several  methods  of  finding  the  specific  heat  of  a  substance : 
it  is  called  the  method  of  mixtures. 

Let  100  grams  of  mercury  at  100°  C.  be  poured  into  100  grams 
of  water  at  0°  C.,  and  suppose  the  temperature  of  the  mixture  to 
be  3.2°  C.  Let  x  =  the  specific  heat  of  mercury.  Now  100  grams 
of  water  has  been  raised  from  0°  to  3.2°,  requiring  for  this  change 
320  calories.  This  heat  has  been  furnished  by  the  mercury  in 
falling  from  100°  to  3.2°.  This  is  equal  then  to  100  x  x  x  96.8. 
Now,  as  no  heat  has  been  lost,  the  calories  received  by  the  water 
are  equal  to  those  given  up  by  the  mercury  or  320  =  9680  x,  from 
which  we  find  x  =  0.033. 

The  specific  heats  of  substances  are  also  found  by  determining 
the  amounts  of  ice  at  0°  C.,  or  32°  F.,  which  they  will  melt  in 
cooling  from  a  given  temperature  to  that  of  melting  ice. 

The  specific  heat  of  a  substance  in  a  liquid  state  is  generally 
greater  than  in  the  solid  form.  The  specific  heats  of  the  more 
perfect  gases  are  nearly  equal  to  that  of  air,  which  is  0.237. 


328  HEAT. 

Ex.  1.  A  coil  of  copper  wire  weighing  45.1  grams  was  dropped 
into  a  calorimeter  containing  52.5  grams  of  water  at  10°.  The 
copper  before  immersion  was  at  99.6°  C.,  and  the  common  tem- 
perature of  copper  and  water  after  immersion  was  16.8°  C.  Find 
the  specific  heat  of  the  copper  wire. 

515.  Apparent  Conduction  Affected  by  Specific  Heat. 
— The  conducting  power  of  different  substances  cannot  be  cor- 
rectly compared,  without  making  allowance  for  their  specific  heat 
(Art.  501).     For  the  heat  which  is  communicated  to  one  end  of  a 
rod,  will  collect  at  the  other  end  more  slowly,  if  a  great  share  of  it 
disappears  on  the  way.     For  instance,  at  the  same  distance  from 
the  source  of  heat,  wax  is  melted  quicker  on  a  rod  of  bismuth  than 
on  one  of  iron,  though  iron  is  the  better  conductor,  because  the 
specific  heat  of  iron  is  three  times  as  great  as  that  of  bismuth ; 
the  heat  actually  reaches  the  wax  soonest  through  the  iron,  but  not 
enough  to  melt  it,  because  so  much  is  required  to  raise  the  iron  to 
a  given  temperature. 

516.  Changes   of  Condition. — Among  the  most  important 
effects  produced  by  heat  are  the  changes  of  condition  from  solid 
to  liquid  and  from  liquid  to  gas,  or  the  reverse,  according  as  the 
temperature   of  a  body  is  raised  or  lowered.     Increase  of  heat 
changes  ice  to  water,  and  water  to  steam,  and  the  diminution  of 
heat  reverses  these  effects.     A  large  part  of  the  simple  substances, 
and  of  compound  ones  not  decomposed  by  heat,  undergo  similar 
changes  at  some  temperature  or  other  ;  and  probably  it  would  be 
found  true  of  all  if  the  requisite  temperature  could  be  reached. 

The  melting-point  (called  also  freezing-point,  or  point  of  conge- 
lation) of  a  substance  is  the  temperature  at  which  it  changes  from 
a  solid  to  a  liquid,  or  the  reverse. 

The  boiling-point  is  the  temperature  at  which  it  changes  from  a 
liquid  to  a  gas,  or  the  reverse. 

517.  Latent  Heat. — Whenever  a  solid  becomes  a  liquid,  or 
a  liquid  becomes  a  gas,  a  large  amount  of  heat  disappears,  and  is 
said  to  become  latent.     The  heat-energy  is  expended  in  sunder- 
ing the  atoms,   and  perhaps  in  putting  them  into  new  relations 
and  combinations,  so  that  there  is  not  the  slightest  increase  of 
temperature  after  the  change  begins  till  it  ends.     The  energy  is. 
not  lost,  but  is  treasured  up  in  the  form  of  potential  energy,  which 
becomes  available  whenever  a  change  is  made  in   the  opposite 
direction.     Using  the  heat-energy  to  turn  water  into  steam,  is  like 
using  the  strength  of  the  ai-ni  in  coiling  up  a  spring,  or  lifting 
a  weight  from  the  earth.     The  spring  and  the  weight  are  each  in 


LATENT    HEAT.  329 

a  condition  to  perform  work.     They  have  potential  energy,  which 
can  be  used  at  pleasure. 

It  has  been  already  noticed   that  much   heat   disappears  in 
bodies  of  great  specific  heat,  as  their  temperature  rises.     But  the 
amount  which  becomes  latent,  while  a  change  of  condition  takes' 
piace,  is  vastly  greater. 

The  number  of  calories  necessary  to  transform  one  gram  of 
water  at  100°  C.  into  steam  at  the  same  temperature  is  called  the 
latent  heat  of  steam,  and  equals  537.  To  transform  n  grams  would, 
of  course,  take  n  times  as  many  calories.  Similarly  the  latent  heat 
of  water  (or  ice)  is  the  number  of  calories  necessary  to  transform 
one  gram  of  ice  at  0°  into  water  at  0°  C.  The  latent  heat  of  ice 
is  80. 

In  steam  boilers  the  pressure  exerted  by  the  steam  upon  the 
surface  of  the  water  has  the  effect  of  raising  the  water's  boiling- 
point.  The  latent  heat  of  the  steam  under  such  conditions  is 
greater  than  the  value  given  above.  The  determination  of  the 
energy  stored  up  in  steam  at  different  temperatures  and  under 
different  pressures  is  a  problem  of  great  technical  importance. 
The  total  number  of  calories,  Q,  required  to  change  n  grams  of 
wrater  at  0°  C.  into  steam  at  t°  C.  is  expressed  by  the  formula 
Q  =  (606.5  +  0.305  t)  n. 

For  100°  C.,  Q  =  637  n;  for  150°  C.,  Q  =  652.2  n;  for  200° 
C.,  Q  =  667.5  H. 

518.  Fusion  or  Melting. — The  change  from  the  solid  to 
the  liquid  state  may  be  either  very  gradual  or  very  abrupt.  As 
the  temperature  rises,  many  substances  become  pasty,  like  wrought 
iron  at  white  heat,  and  for  a  considerable  range  of  temperatures 
such  substances  are  neither  solid  nor  liquid,  and  no  definite  melt- 
ing-point can  be  assigned.  Ice  passes  very  abruptly  from  the 
solid  to  the  liquid  state,  probably  during  a  rise  of  temperature  not 
greater  than  0.1°  C. 

From  the  beginning  of  fusion  till  the  end  of  the  change  of  con- 
dition there  is  no  rise  of  temperature,  the  heat  which  does  internal 
work  being  termed  latent  heat  of  fusion. 

The  latent  heat  of  fusion  of  ice  has  already  been  given  (Art. 
517).  The  melting-points  of  a  few  substances  are  given  below : 


Ice O'C. 

Spermaceti 49'  C. 

White  Wax  . .  .     65"  C. 


Sulphur....; 111°  C. 


Tin 235°  C. 

Lead 325°  C. 

Silver  ...  . .   1000°  C. 


Iron 1500°  C. 


The  melting-point  of  a  substance  which  expands  on  solidify- 
ing is  lowered  by  great  increase  of  pressure  above  the  ordinary 


330  HEAT. 

pressure  of  the  atmosphere,  while  that  of  a  substance  which  con- 
tracts in  solidifying  is  raised.  The  melting  point  of  wax  was 
raised  from  65°  C.  to  75°  C.  by  a  pressure  of  520  atmospheres, 
.while  the  melting  point  of  ice  is  lowered  about  0.0074°  C.  for 
every  additional  pressure  of  one  atmosphere. 

Alloys  are  generally  more  fusible  than  the  metals  of  which 
they  are  composed. 

519.  Vaporization. — The   change  from  the  liquid  to  the 
gaseous  state  is  termed  vaporization.     This  change  is  sometimes 
effected  quietly  without  the  formation  of  bubbles,  then  termed 
evaporation,  and  sometimes  in  a  violent  manner  with  the  forma- 
tion of  bubbles,  to  which  action  the  term  ebullition  is  applied. 

Vaporization  is  more  rapid  as  the  pressure  upon  the  surface 
of  the  liquid  is  diminished. 

The  boiling  point  of  water  at  one  atmosphere,  at  the  level  of 
the  ocean,  is  100°  C.;  but  upon  the  tops  of  high  mountains  the 
boiling  point  is  90°  and  85°  C.,  and  in  the  air  pump  vacuum  it  is 
as  low  as  23°  C. 

The  effect  of  diminished  pressure  to  lower  the  boiling  point  is 
well  shown  by  the  following  familiar  experiment :  In  a  thin  glass 
flask,  boil  a  little  water,  and  after  removing  it  from  the  fire,  cork 
and  invert  the  flask.  The  steam  which  is  formed  will  soon  press 
so  strongly  upon  the  water  as  to  stop  the  boiling.  When  this 
happens,  pour  a  little  cold  water  upon  the  flask ;  the  water  within 
will  immediately  commence  boiling  violently,  because  the  vapor 
is  condensed  and  the  pressure  removed.  This  effect  may  be 
reproduced  several  times  before  the  water  in  the  flask  is  too  cool 
to  boil  in  a  vacuum. 

520.  Other  Causes  Affecting  the  Boiling  Point.— The 

boiling  point  is  raised  by  substances  in  solution,  provided  they  are 
less  volatile  than  the  liquid  in  which  they  are  dissolved. 

Water  saturated  with  common  salt  boils  at  109°  C.,  and  when 
chloride  of  calcium  replaces  the  salt  the  boiling  point  is  raised  to 
179°  C.  Substances  held  in  suspension,  but  not  dissolved,  have 
no  effect  upon  the  boiling  point. 

Water  from  which  the  dissolved  air  has  been  removed  by 
previous  ebullition,  has  been  raised  to  112°  C.  before  boiling,  the 
elastic  air  seeming  to  act  as  a  spring  to  aid  ebullition. 

Water  boils  at  a  higher  temperature  in  glass  vessels  than  in 
metallic  ones,  rising  as  high  as  105°  C.  before  ebullition  begins. 
If  metal  clippings  or  filings,  or  any  angular  fragments  whatever 
which  may  serve  as  a  nucleus,  be  dropped  into  the  flask,  the  boiling 
point  is  brought  down  to  100°  C.,  and  the  violent  bumping  which 
accompanies  ebullition  at  the  higher  temperatures  is  prevented. 


EVAPOBATION.  33  j_ 

521.  Spheroidal  Condition. — When  a  little  water  is  placed 
in  a  red-hot  metallic  cup,  instead  of  boiling  violently,  and  disap- 
pearing in  a  moment,  as  might  be  expected,  it  rolls  about  quietly 
in  the  shape  of  an  oblate  spheroid,  and  wastes  very  slowly.  So 
drops  of  water,  falling  on  the  horizontal  surface  of  a  very  hot 
stove,  are  not  thrown  off  in  steam  and  spray  with  a  loud  hissing 
sound,  as  they  are  when  the  stove  is  only  moderately  heated,  but 
roll  over  the  surface  in  balls,  slowly  diminishing  in  size  till  they 
disappear. 

In  such  cases,  the  water  is  said  to  be  in  the  spheroidal  state. 
Not  being  in  contact  with  the  metal,  it  assumes  the  shape  of  an 
oblate  spheroid,  in  obedience  to  its  own  molecular  attractions  and 
the  force  of  gravity,  as  small  masses  of  mercury  do  on  a  table. 
The  reason  why  the  water  does  not  toucli  the  hot  metal  is,  that 
the  heat  causes  a  coat  of  vapor  to  be  instantly  formed  about  the 
drop,  on  which  it  rests  as  on  an  elastic  cushion  ;  and  as  the  vapor 
is  a  poor  conductor  of  heat,  further  evaporation  proceeds  very 
slowly.  It  is  easily  seen  that  the  spheroid  does  not  touch  the 
metal,  by  so  arranging  the  experiment  that  a  beam  of  light  may 
shine  horizontally  upon  the  drop,  and  cast  its  shadow  completely 
separated  from  that  of  the  hot  plate  below  it,  as  in  Fig.  302. 

FIG.  302. 


If  the  heated  surface  is  cooling,  the  temperature  may  become 
so  low  that  the  drop  at  length  touches  it,  when  in  an  instant 
violent  ebullition  takes  place,  and  the  water  quickly  disappears  in 
vapor. 

522.  Evaporation. — Many  liquids  and  even  solids  pass  into 
the  gaseous  state  by  a  slow  and  almost  insensible  process,  which 
goes  on  at  the  surface.     This  is  called  evaporation  ;  and  it  takes 
place  at  all  temperatures,  but  more  rapidly  as  the  temperature  is 
higher.     Ice  and  snow  waste  aAvay  gradually  at  temperatures  far 
below  0°  C.,  and  the  odor  of  brass,  copper,  and  iron  is  attributed 
to  an  insensible  evaporation  of  these  metals. 

523.  Condensation. — The  change  from  the  condition  of 
vapor  to  the  liquid  state  is  called  condensation.     This  change  of 
state  may  be  caused  by  cooling  and  by  compression.     A  saturated 


332  HEAT. 

vapor  at  any  given  temperature  and  pressure  will  be  partially  con- 
densed by  either  lowering  the  temperature  or  by  increasing  the 
pressure.  Those  gases  which  have  usually  been  called  permanent 
gases,  because,  under  ordinary  conditions  they  are  very  far  re- 
moved from  their  point  of  condensation,  have  been  reduced  to 
the  liquid  state  by  very  low  temperatures  and  great  pressures 
combined. 

Vapors  give  up  their  latent  heat  of  vaporization  during  the 
process  of  condensation  ;  the  latent  heat  of  steam  may  be  deter- 
mined by  passing  a  known  weight  of  steam  at  100°  C.  into  a 
given  quantity  of  water  at  a  known  temperature,  and  taking  the 
resulting  temperature. 

Suppose  10  grams  of  steam  at  100°  C.  to  be  condensed  by  60 
grams  of  water  at  0°  C.,  and  that  the  resulting  temperature  is  91°  C. 
The  60  grams  of  water  raised  from  0°  to  91°  required  60  x  91  =  5460 
heat  units ;  10  grams  of  steam,  after  condensation  at  100°,  gave  up 
90  of  these  5460  units  in  cooling  from  100°  to  91°,  leaving  537  to 
each  gram  of  steam  as  the  latent  heat  of  its  vaporization. 

524.  Solidification.— Substances  which  have  been  melted 
and  which  cool  slowly  while  passing  into  the  solid  state  usually 
assume  a  regular  crystalline  structure.    If  they  expand  on  solidify- 
ing, the  solid  will  float  in  the  liquid,  but  if  they  contract  the 
solid  will  sink. 

A  liquid  may  be  cooled  below  its  normal  temperature  of  solidi- 
fication. A  hot  saturated  solution  of  Glauber's  salt,  cooled  slowly 
and  at  rest,  will  remain  liquid  at  the  ordinary  temperature  of  the 
atmosphere ;  but  upon  being  suddenly  jarred,  or  when  a  crystal 
of  the  salt  is  dropped  into  the  liquid,  the  molecular  •equilibrium 
is  destroyed  and  solidification  ensues  at  once.  Water  which  has 
been  boiled,  to  free  it  from  air,  may  be  cooled  to  — 10°  C.,  or 
even  lower,  without  freezing;  but  any  vibration  causes  instant 
crystallization.  In  all  such  cases  the  latent  heat  of  fusion  becomes 
sensible,  and  may  be  felt  by  placing  the  hands  upon  the  contain- 
ing vessel. 

The  freezing  point  of  water  containing  salt  in  solution  is 
lower  than  that  of  pure  water.  Sea  water  freezes  at  — 2.5°  0.  to 
— 3°  C.  ;  the  ice  is  pure,  containing  none  of  the  salt. 

525.  Freezing    Produced   by    Melting.— Since  a  great 
amount  of  heat  disappears  in  a  substance  as  it  passes  from  the 
solid  to  the  liquid  state,  the  loss  thus  occasioned  may  produce 
freezing  in  a  contiguous  body.     When  salt  and  powdered  ice  are 
mixed,  their  union  causes  liquefaction.     And  if  this  mixture  is 
surrounded  by  bad  conductors,  and  a  tin  vessel  containing  some 


FREEZING     BY    EVAPORATION.  333 

liquid  be  placed  in  the  midst  of  it,  the  latter  is  frozen  by  the 
abstraction  of  heat  from  it,  by  the  melting  of  the  ice  and  salt.  In 
this  way  ice  creams  and  similar  luxuries  are  easily  prepared  in 
hot  as  well  as  in  cold  weather. 

526.  Freezing  by  Evaporation. — In  like  manner,  freezing 
by  evaporation  is  explained.     Put  a  little  water  in  a  shallow  dish 
of  thin  glass,   and  set  it  on  a  slender  wire-support  under  the 
receiver  of  an  air  pump.     Beneath  the  wire-support  place  a  broad 
dish  containing  sulphuric  acid.    When  the  air  is  exhausted,  the 
water  in  a  few  moments  is  found  frozen.     As  the  pressure  of  the 
air  is  taken  off,  evaporation  proceeds  with  increased  rapidity,  and 
the  requisite  heat  for  this  change  of  condition  can  be  taken  only 
from  the  dish  of  water.     But  the  atmosphere  of  vapor  retards  the 
process  by  its  pressure  ;  hence  the  sulphuric  acid  is  placed  in  the 
receiver,  so  as  to  seize  upon  the  vapor  as  fast  as  formed,  and  thus 
render  the  vacuum  more  complete.    The  water  is  frozen  by  giving 
up  its  heat  to  become  latent  in  the  vapor,  so  rapidly  formed  ;  but 
when  this  vapor  becomes  liquid  again  in  combining  with  the  acid,, 
the  same  heat  reappears  in  raising  the  temperature  of  the  acid. 

Thin  cakes  of  ice  may  sometimes  be  procured,  even  in  the 
hottest  climates,  by  the  evaporation  of  water  in  broad  shallow- 
pans  under  the  open  sky,  where  radiation  by  night  aids  in  redu- 
cing the  temperature.  The  pans  should  be  so  situated  as  to  receive 
the  least  possible  heat  by  conduction. 

Various  ice-making  machines  have  been  devised  in  which  the 
vaporization  of  some  volatile  liquid,  such  as  ether,  liquid  am- 
monia, liquid  sulphurous  acid,  &c.,  abstracts  sufficient  heat  from 
the  water  to  freeze  it. 

527.  Regelation.— If  two  pieces  of  ice  at  0°  C.,  having 
smooth  surfaces,  be  pressed  together,  they  will  soon  adhere,  and 
will  do  this  in  air,  in  water,  or  in  vacuo.     This  freezing  together 
again  is  called  regelation. 

The  interior  of  a  block  of  melting  ice  is  a  little  colder  than  the 
surface  :  now  when  the  two  surfaces  are  pressed  together,  the  very 
thin  film  of  water  which  covers  them  is  removed  from  the  warmer 
air,  and  is  in  the  same  condition  as  though  transferred  to  the 
interior  of  a  block,  the  lower  temperature  of  which  freezes  it. 


334 


HEAT 


CHAPTER    IV. 

TENSION    OF  VAPOR.— THE    STEAM-ENGINE.-MECHANICAL 
EQUIVALENT    OF  HEAT. 

528.  Dalton's  Laws.— 

1.  Whatever  be  the  temperature  of  a  liquid  which  partly  fills 
a  vessel,  vaporization  will  go  on  till  the  vessel  is  filled  with  vapor, 
of  a  density  determined  solely  by  the  temperature,  after  which 
vaporization  will  cease. 

2.  If  the  space  occupied  by  the  vapor  be  made  larger,  the  tem- 
perature being  the  same,  then  vaporization  will  again  go  on  till 
the  density  is  the  same  as  before.     If  the  space  be  made  smaller, 
the  temperature  remaining  constant,  a  part  of  the  vapor  returns 
to  the  liquid  state,  and  the  remaining  vapor  will  have  the  same 
density  as  before. 

3.  If,  besides  the  liquid  and  its  vapor,  the  vessel  contains  any 
gas,  not  capable  of  chemical  action  on  the  .liquid,  then  exactly  the 
same  amount  of  vapor,  of  the  same  density  as  before,  will   be 
formed ;  but  the  time  required  to  reach  the  maximum  density 

will  be  greater  because  of  the  mechanical  obstruc- 
tion to  a  rapid  diffusion,  which  the  gas  offers. 

A  vapor  at  the  maximum  density  and  pressure 
for  the  given  temperature  is  called  a  saturated 
vapor. 

529.  Experimental  Illustration. — Fill  a 
barometer  tube  A  B  (Fig.  303)  full  of  mercury ; 
close  the  open  end  with  the  finger  and  invert  into 
the  cup  H  of  the  deep  mercury  cistern  H  K. 
With  a  pipette,  the  tube  of  which  is  bent  upwards 
at  the  end,  transfer  enough  ether  to  the  barometer 
tube  to  leave  a  thin  film  of  liquid  c  d,  after  the 
space  A  d  is  filled  with  saturated  vapor.  Measure 
the  height  c  H  of  the  mercury  column.  If  the 
tube  A  B  be  raised,  tending  to  increase  the  space 
A  d  above  the  liquid,  more  vapor  will  form  and 
c  H  will  remain  unaltered  ;  if  the  tube  A  B  be 
depressed,  tending  to  diminish  the  space  A  d, 
vapor  will  condense  to  liquid  again,  and  c  H  will 
still  be  unaltered. 

To  show  the  effect  of  change  of  temperature  use  a  barometer 


FIG.  303. 


TENSIONS    OF     VAPORS. 


335 


FIG.  804. 


tube  bent  at  its  closed  end  as  in  Fig.  304,  so  that  a  portion  of  the 
bend  may  either  be  surrounded  with  cooling  mixtures,  as  at  A, 
or  may  be  warmed  by  a  flame.  Upon  raising  the  temperature  of 
the  contained  vapor  its  ten- 
sion will  increase  and  the 
mercury  column  C  B  will  be 
shortened ;  upon  lowering 
the  temperature  the  tension 
will  decrease,  and  C  B  will 


FIG.  S05. 
B   C 


lengthen     as    the     mercury 

rises. 

530.  Tensions  of  Dif- 
ferent Vapors.  —  Transfer 
to  three  barometer  tubes,  A, 
B,  and  C  (Fig.  305),  water, 
alcohol  and  ether  respectively, 
and  the  mercury  columns 
will  stand  at  different  heights 
a,  b  and  c,  showing  that  the 
tensions  of  the  three  vapors 
are  not  the  same  at  the  same 
temperature. 

631.  Tension  in  Gen- 
erator and  Condenser.— 

Let  two  vessels,  A  and  B  (Fig.  306),  be  connected  by  a  pipe 
furnished  with  a  stop-cock  C.  Let  the  tubes  a  and  b  be  connected 
with  separate  manometers  to  indicate  the  tensions  of  the  vapor  in 
A  and  B  when  C  is  closed.  Having  partly  filled  A  with  water, 

cause  it  to  boil  until  all  air 
has  been  driven  from  both 
flasks  through  C  and  the 
loosened  stopper  of  B ;  now 
close  B  and  remove  the  lamp. 
The  two  manometers  will 
indicate  the  same  tension  in 
both  flasks. 

Now  close  C'and  surround 
B  with  cold  water;  part  of 
the  vapor  in  it  will  be  con- 
densed, the  remainder  hav- 
ing a  greatly  reduced  tension  as  shown  by  the  fall  of  the  mercury 
column  in  its  manometer.  Apply  heat  to  A,  thus  forming  nev 
vapor  of  higher  tension  than  before,  as  will  be  shown  by  its 
manometer  reading. 


FIG. 


336  HEAT. 

If  now  C  be  opened  the  manometer  connected  with  A  will  fall 
to  the  same  reading  as  that  of  B,  and  the  two  will  indicate  this 
same  reading  just  as  long  as  the  temperature  of  B  is  kept  con- 
stant and  below  that  of  A. 

The  liquid  in  A  merely  distills  over  to  B,  at  the  tension  of  the 
vapor  in  the  colder  vessel. 

532.  Heat  Energy  in  Steam. — It  has  been  already  noticed 
that  while  water  is  heated,  and  especially  after  it  is  converted  into 
steam  by  boiling,   the  heat   apparently  lost  is  so  much  energy 
treasured  up  ready  for  use,  as  truly  as  when  energy  is  expended 
in  lifting  great  weights,  which  by  their  descent  can  do  the  work 
desired.     In  modern    engineering,   the   energy  of  steam   is   em- 
ployed more  extensively,  and  for  more  varied  purposes,  than  any 
other.     Every  steam-engine   is  a   machine  for  transforming   the 
internal  motion  of  heated  steam  into  some  of  the  visible  forms  of 
motion. 

533.  Tension  of  Steam. — When  steam  is  formed  by  boiling 
•water  in  the  open  air,  its  tension  is  equal  to  that  of  the  air,  and 
therefore  ordinarily  about  fifteen  pounds  to  the  square  inch.     But 
when  it  is  formed  in  a  tight  vessel,  so  that  it  cannot  expand,  as 
the  temperature  of  the  water  is  raised  the  tension  is  increased  in 
n  much  greater  ratio ;  because  the  same  steam  has  greater  tension 
at  a  higher  temperature,  and  besides  this,  new  steam  is  continually 
-added. 

The  following  table  gives  the  temperature  corresponding  to 
various  atmospheres  of  tension  : 

Atmospheres,  1245  9          10        14        15         19      20 

Temperature  C.°,  100  120.6  144  152.2  175.8   180.3  195.5  198.8  210.4  213 

534.  The  Steam-Engines  of  Savery  and  Newcomen. — 
The  only  steam-engines  that   were  at  all  successful   before   the 
great  improvements  made  by  Watt,  were  the  engine  of  Savery  and 
that  of  Newcomen.     No  other  purpose  was  proposed  by  either 
than  that  of  removing  water  from  mines. 

In  the  engine  of  Savery,  steam  was  made  to  raise  water  by 
acting  on  it  directly,  and  not  through  the  intervention  of  ma- 
chinery. 

It  consisted  of  a  boiler  E  (Fig.  307) ;  a  cylinder  A,  with  a 
valve  at  c  opening  inward,  and  one  at  d  opening  outward ;  a  pipe 
e  to  discharge  cold  water  upon  the  cylinder,  and  a  steam  pipe  /, 
from  the  boiler  to  the  cylinder. 

First  the  steam-cock  at  /  is  opened  and  steam  fills  the  cylinder 
A,  driving  the  air  out  through  the  valve  d.  Next  f  is  closed  and 


STEAM     ENGINES. 


337 


the  cock  e  is  opened,  allowing  cold  water  to  flow  over  the  cylinder 
from  the  delivery  pipe  0,  thus  condensing  the  steam  in  A,  and 
creating  a  vacuum,  into  which  the  atmosphere  forces  water  from 
the  supply  P,  through  the  valve  c.  Now  e  is  closed  and  f  opened 

FIG.  307.  FIG.  308. 

O 


again,  and  steam  enters  the  cylinder  A  and  drives  the  water  out 
through  d.  When  A  is  full  of  steam  the  operation  is  repeated  as 
"before. 

Newcomen  used  steam  to  work  a  common  pump.  The  weighted 
pump  rod  R  (Fig.  308)  was  attached  to  one  end  of  a  working 
beam  B,  while  at  the  other  end  of  B  was  hung  the  piston  P, 
working  steam  tight  in  the  cylinder  C.  Steam  at  atmospheric 
pressure  from  the  boiler  G  enters  C  through  the  cock  a,  and  P 
being  pressed  upon  equally  on  both  sides  is  drawn  to  the  top  of  C 
"by  the  weight  of  the  pump  rod  R.  Now  a  is  closed  and  e  is 
opened,  permitting  cold  water  from  a  tank  T  to  flow  into  C, 
which  condenses  the  steam,  creating  a  vacuum,  and  allows  the 
piston  to  descend  under  atmospheric  pressure.  When  P  has 
reached  the  bottom  of  C,  e  having  been  closed,  a  and  d  are  opened 
and  steam  enters  the  cylinder,  while  the  injection  water  from  T 
flows  out  through  d,  and  the  piston  P  rises  as  at  first.  On  closing 
a  and  d  and  opening  e  the  stroke  is  repeated. 

As  the  water  was  raised  by  the  direct  pressure  of  the  atmos- 
phere, this  invention  of  Newcomen  was  called  the  atmospheric 
engine. 

In  these  diagrams  of  Savery's  and  Newcomen's  engines,  all 
details  of  valves  or  other  working  parts  have  been  omitted,  that 
the  principle  alone  might  claim  attention. 


338 


HEAT. 


In  neither  of  these  methods  was  steam  used  economically  as  a 
power.  The  movements  in  both  cases  were  sluggish,  and  a  large 
part  of  the  energy  was  wasted,  because  the  steam  was  compelled 
to  act  upon  a  cold  surface,  which  condensed  it  before  its  work 
was  done. 

535.  The  Steam-Engine  of  Watt. — Steam  did  not  give 
promise  of  being  essentially  useful  as  a  power  till  "Watt,  in  the 
year  1760,  made  a  change  in  the  atmospheric  engine,  which  pre- 
vented the  great  waste  of  force.     Newcomen  introduced  the  cold 
water  which  was  to  condense  the  steam  into  the  steam  cylinder 
itself ;  and  the  cylinder  must  be  cooled  to  a  temperature  below 
100°  F.,  else  there  would  be  steam  of  low  tension  to  retard  the 
descent  of  the  piston.     But  when  the  piston  was  to  be  raised, 
the  cylinder  must  be  heated  again  to  212°  F.,  in  order  that  the 
admitted  steam  might  balance  the  pressure  of  the  air. 

In  the  engine  of  Watt,  the  steam  is  condensed  in  a  separate 
vessel  called  the  condenser.  The  steam  cylinder  is  thus  kept  at 
the  uniform  temperature  of  the  steam.  In  the  first  form  which 
he  gave  to  his  engine,  he  so  far  copied  the  atmospheric  engine  as 
to  allow  the  piston,  after  being  pressed  down  by  steam,  to  be  raised 
again  by  the  load  on  the  opposite  end  of  the  great  beam,  while  the 
steam  circulates  freely  below  and  above  the  piston.  This  was 
called  the  single-acting  engine, 
and  might  be  successfully  used 
for  the  only  purpose  to  which  any 
steam-engine  was  as  yet  applied, 
namely,  pumping  water  from 
mines.  But  he  almost  imme- 
diately introduced  the  change  by 
which  the  whole  force  of  the 
steam  was  brought  to  act  on  the 
upper  and  the  under  side  of  the 
piston.  It  thus  became  double- 
acting,  and  the  steam  force  was 
no  longer  intermittent. 

536.  The    Double-acting 
Engine.— Let  8  (Fig.  309)  be 
the  steam  cylinder,  P  the  piston, 
A  the  piston  rod,  passing  with 
steam-tight  joint  through  the  top 
of  the  cylinder,  C  the  condenser, 
kept  cold  by  the  water  of  the 

cistern  G,   B  the  steam  pipe  from  the  boiler,  K  the  eduction 
pipe,  which  opens  into  the  valve  chest  at  0,  D  D  the  D-valve, 


FIG  .  309. 


CONDENSING     ENGINE.  339 

E  F  the  openings  from  the  valve-chest  into  the  cylinder.  As  the 
D-valve  is  situated  in  the  figure,  the  steam  can  pass  through  B 
and  E  into  the  cylinder  below  the  piston,  while  the  steam  above 
the  piston  can  escape  by  F  through  0  and  K  to  the  condenser, 
where  it  is  condensed  as  fast  as  it  enters ;  so  that  in  an  instant 
the  space  above  the  piston  is  a  vacuum,  while  the  whole  force  of 
the  steam  is  exerted  on  the  under  side.  The  piston  is  therefore 
driven  upward  without  any  force  to  oppose  it.  But  before  it 
reaches  the  top,  the  D-valve,  moved  by  the  machinery,  begins  to 
descend,  and  shut  off  the  steam  from  E  and  admit  it  to  F,  and, 
on  the  other  hand,  to  shut  F  from  the  eduction  pipe  0,  and  open 
E  to  the  same.  The  steam  will  then  press  on  the  top  of  the 
piston,  and  there  will  be  a  vacuum  below  it,  .so  that  the  piston 
descends  with  the  whole  force  of  the  steam,  and  without  resist- 
ance. To  render  the  condensation  more  sudden,  a  little  cold 
water  is  thrown  into  the  condenser  at  each  stroke  through  the 
pipe  H. 

537.  Condensing  Engine. — The  principle  of  the  condens- 
ing engine  is  illustrated  by  the  figure  and  description  of  the 
preceding  article.  But  the  condensing  apparatus  of  this  kind  of 
engine  requires  many  other  parts,  most  of  which  are  presented  in 
Fig.  310.  C  is  the  steam  cylinder;  R  the  rod  connecting  its 

FIG.  310 


piston  with  the  end  of  the  working  beam,  not  represented  ;  A  the 
steam-pipe  and  throttle-valve ;  B  B  the  D-valve ;  D  D  the  educ- 
tion pipe,  leading  from  the  valve-chest  to  the  condenser  E\  GO 


340  HEAT. 

the  cold  water  surrounding  the  condenser ;  F  the  air-pump,  which 
keeps  the  condenser  clear  of  air,  steam,  and  water  of  condensa- 
tion ;  /  the  hot  well,  in  which  the  water  of  condensation  is  de- 
posited by  the  air-pump ;  K  the  hot-water  pump,  which  forces 
the  water  in  the  hot  well  through  L  to  the  boiler ;  H  the  cold- 
water  pump,  by  which  water  is  brought  to  the  cistern  G  0 ;  the 
rods  of  all  the  pumps,  F,  K,  and  H,  are  moved  by  the  working 
beam ;  P  the  fly-wheel ;  M  the  crank  of  the  same,  N  the  connect- 
ing-rod, by  which  the  working  beam  conveys  motion  to  the  fly- 
wheel ;  Q  the  excentric  rod,  by  which  the  D-valve  is  moved  ;  0  0 
the  governor,  which  regulates  the  throttle- valve  in  the  steam- 
pipe  A. 

There  is  much  economy  of  fuel  and  saving  of  wear  in  the 
machinery,  arising  from  the  proper  adjustment  of  the  valves.  If 
the  steam  enters  the  cylinder  during  the  whole  length  of  a  stroke 
of  the  piston,  its  motion  is  accelerated ;  and  is  therefore  swiftest 
at  the  instant  before  being  stopped  ;  thus  the  machinery  receives 
a  violent  shock.  If  the  valve  is  adjusted  to  cut  off  the  steam 
when  the  piston  has  made  one-third  or  one-half  of  its  stroke,  the 
diminishing  tension  may  exert  about  force  enough,  during  the 
remaining  part,  to  keep  up  a  uniform  motion.  The  cut-off,  how- 
ever, should  be  regulated  in  each  engine,  according  to  friction 
and  other  obstructions. 

538.  Non-condensing    Engine. —  For    many    purposes, 
especially  those  of  locomotion,  it  is  advantageous  to  dispense  with 
the  large  weight  and  bulk  of  machinery  necessary  for  condensa- 
tion, and  do  the  work  with  steam  of  a  higher  tension.     If  (Fig. 
310)  the  condenser,  cistern,  and  all  the  pumps  are  removed,  then 
the  steam  is  discharged  from  E  and  F  at  each  stroke  into  the  air. 
Therefore  the  steam  in  that  part  of  the  cylinder  which  is  open  to 
the  air,  will  have  a  tension  of  15  Ibs.  per  inch  ;  and,  consequently^ 
the  steam  on  the  opposite  side  of  the  piston  must  have  a  ten- 
sion 15  Ibs.  per  inch  greater   than   before,  in   order  to  do  the 
same  work. 

Steam  of  a  pressure  not  greater  than  45  Ibs.  per  inch  (above  the 
atmosphere)  is  called  low  pressure  steam,  or  low  steam  ;  high  steam 
is  at  a  pressure  above  this,  and  not  uncommonly  runs  higher  than 
200  Ibs.  per  inch  by  the  gauge. 

539.  Calculation  of  Steam  Power.  —  Assuming  that  the 
pressure  within  the  cylinder  of  a  steam-engine  remains  constant 
throughout  the  whole  of  the  stroke,  we  can  find  the  horse-power 
developed  in  each  cylinder  of  an  engine,  having  given — 

A  —  area  of  pis'ton  in  square  inches. 


MECHANICAL     EQUIVALENT     OF     HEAT.  34-1 


P  =•  pressure  upon  the  piston  in  pounds  per  square  inch. 

S  =  length  of  stroke  in  feet. 

E  =  number  of  revolutions  per  minute. 

Here  P  denotes  the  intensity  of  the  pressure  on  the  piston  in 
pounds  weight  per  square  inch.  The  total  pressure  on  the  piston 
is  the  weight  of  A  P  pounds.  This  is  the  acting  force,  and  the 
distance  through  which  it  moves  in  each  stroke  is  S  feet. 

Thus  the  work  done  in  each  stroke  is  S  A  P  foot-pounds. 

Since  there  are  two  strokes  for  each  revolution,  the  number  of 
strokes  per  minute  is  2  R,  and  the  work  done  per  minute  is  2  SR  A  P 
foot-pounds.     Thus  the  horse-power  developed  is 
H.  P.  =  2  SJ24  P/33,000. 

540.  Mechanical  Equivalent  of  Heat. — We  have  seen  that 
a  given  quantity  of  heat  represents  a  definite  amount  of  energy,  and 
accordingly  a  gram  calorie  must  be  equivalent  to  a  certain  number 
of  ergs  (Art.  33).  Joule,  Rowland,  and  others  have  performed 
experiments  which  demanded  the  utmost  accuracy  and  patience, 
and  have  determined  that  one  calorie  is  equivalent  to  416  x  10* 
ergs.  This  coefficient  is  called  the  mechanical  equivalent  of  heat. 
Joule,  who  made  the  first  determination,  used  the  foot-pound  and 
the  pound-degree-Fahrenheit  as  units  of  work  and  heat  respectively. 
The  mechanical  equivalent  in  these  units  may  be  expressed  by  the 
following  statement : 

The  energy  required  to  heat  one  pound  of  water  one  degree  F.,  is 
equal  to  that  which  would  lift  773  pounds  the  vertical  distance  of  one 
foot,  or  is  equal  to  77 '2  foot-pounds. 

Joule's  mode  of  determining  this  value  of  the  mechanical 
equivalent  is  the  following  : 

A  weight  W  (Fig.  311),  by  means  of  a  cord  passing  over 
a  pully  p  and  around  a  drum  D, 
gives  to  the  vertical  axis  A  a  rapid 
rotation.  Attached  to  this  axis 
are  a  number  of  radial  arms,  or 
paddles,  as  shown  in  the  figure  ; 
projecting  from  the  sides  of  the 
cylinder  C,  in  which  these  arms 
rotate,  are  fixed  anus,  as  shown,  to 
arrest  any  tendency  to  a  rotary  mo- 
tion of  the  water  in  the  cylinder. 

If  one  pound  of  water  at  60°  F. 
be  put  into  the  cylinder  C,  it  will 
require  the  expenditure  of  772  foot- 
pounds of  energy  on  the  part  of 
o  falling  weight  W  to  raise  its  temperature  by  agitation  to  61°  F. 


FIG.  311. 


342  HEAT. 

Heat  is  the  lowest  form  of  energy.  When  by  any  action  energy- 
is  liberated  which  is  not  specially  directed  by  circumstances,  it 
takes  the  form  of  heat.  The  energy  of  a  dynamo-electric  machine, 
if  not  made  to  do  outside  work,  transforms  itself  into  heat  in  the 
circuit.  All  energy  tends  to  assume  the  form  of  heat,  and  it  may 
be  expected  that  the  total  energy  of  the  universe  will  ultimately 
turn  into  heat,  as  was  stated  in  Art.  37. 


CHAPTER   V. 


TEMPERATURE   OF    THE   ATMOSPHERE.  —  MOISTURE  OF  THE 
ATMOSPHERE.— DRAUGHT  AND   VENTILATION. 


541.  Manner  in  which  the  Air  is  Warmed. — The  space 
through  which  the  earth  moves  around  the  sun  is  intensely  cold, 
probably  75°  below  zero ;  and  the  one  or  two  hundred  miles  of 
height  occupied   by  the  atmosphere  is  too  cold   for   animal  or 
vegetable  life,  except  the  lowest  stratum,  three  or  four  miles  in 
thickness.     This  portion  receives  its  heat  mainly  by  convection. 
The  radiated  heat  of  the  sun  passes  through  the  air,  warming 
it  but  little,  and  on  reaching  the  earth  is  partly  absorbed  by  it. 
The   air  lying   in   contact   with   the   earth,   and   thus   becoming 
warmed,  grows  lighter  and  rises,  while  colder  portions  descend 
and  are  warmed  in  their  turn.     So  long  as  the  sun  is  shining 
on  a  given  region  of  the  earth,  this  circulation  is  going  on  con- 
tinually. 

542.  Limit  of  Perpetual  Frost.  —  At  a  moderate  eleva- 
tion, even  in  the  hottest  climate,  the  temperature  of  the  air  is 
always  as  low  as  the  freezing-point.     Hence  the  permanent  snow 
on  the  higher  mountains  in  all  climates.     The  limit  at  the  equa- 
tor is  about  three  miles  high,  and,  with  many  local  exceptions,  it 
descends  each  way  to  the  polar  regions,  where  it  is  very  near  the 
earth.     The  descent  is  more  rapid  in  the  temperate  than  in  the 
torrid  or  frigid  zones. 

543.  Isothermal  Lines. — These  are  imaginary  lines  on  each 
hemisphere,  through  all  those  points  whose  mean  annual  tempera- 
ture is  the  same.     At  the  equator,  the  mean  temperature  is  about 
82 3  F.,  and  it  decreases  each  way  toward  the  poles,  but  not  equally 


TEMPERATURE     AND     TENSION     OF     V  A  P  O  It .     343 

on  all  meridians.  Hence  the  isothermal  lines  deviate  widely  from 
parallels  of  latitude.  Their  irregularities  are  due  to  the  difference 
between  land  and  water,  in  absorbing  and  communicating  heat, 
to  the  various  elevations  of  land,  especially  ranges  of  mountains, 
to  ocean  currents,  &c.  In  the  northern  hemisphere,  the  isother- 
mal lines,  in  passing  westward  round  the  earth,  generally  descend 
toward  the  equator  in  crossing  the  oceans,  and  ascend  again  in 
crossing  the  continents.  For  example,  the  isothermal  of  50°  R, 
which  passes  through  China  on  the  parallel  of  44°,  ascends  in 
crossing  the  eastern  continent,  and  strikes  Brussels,  lat.  51°  ;  and 
then  on  the  Atlantic,  descends  to  Boston,  lat.  42°,  whence  it  once 
more  ascends  to  the  IS".  W.  coast  of  America.  The  lowest  mean 
temperature  in  the  northern  hemisphere  is  not  far  from  zero,  but 
it  is  not  situated  at  the  north  pole.  Instead  of  this,  there  are  two 
poles  of  greatest  cold,  one  on  the  eastern  continent,  the  other  on 
the  western,  near  20°  from  the  geographical  pole.  There  are  in- 
dications, also,  of  two  south  poles  of  maximum  cold. 

544.  Moisture  of  the  Atmosphere.— By  the  heat  of  the 
sun  all  the  waters  of  the  earth  form  above  them  an  atmosphere  of 
vapor,  or  invisible  moisture,  having  more  or  less  extent  and  ten- 
sion, according  to  several  circumstances.     Even  ice  and  snow,  at 
the  lowest  temperatures,  throw  off  some  vapor. 

At  a  given  temperature,  there  can  exist  an  atmosphere  of 
vapor  of  the  same  height  and  tension,  whether  there  is  an  atmos- 
phere of  oxygen  and  nitrogen  or  not  (Art.  528).  Vapor  is  not 
suspended  in  the  air,  or  dissolved  by  it,  but  exists  independently. 
And  yet  it  is  by  no  means  always  true  that  there  is  actually  the 
same  tension  of  vapor  as  there  would  be  if  it  existed  alone,  because 
of  the  time  required  for  the  formation  of  vapor,  on  account  of 
mechanical  obstruction  presented  by  the  air  ;  whereas,  if  no  air 
existed,  the  vapor  would  form  almost  instantly. 

545.  Temperature  and  Tension  of  Vapor. — The  degree 
of  tension  of  vapor  forming  without  obstruction,  depends  on  its 
temperature,  but  varies  far  more  rapidly,  increasing  pretty  nearly 
in  a  geometrical  ratio,   while  the  heat  increases  arithmetically  ; 
thus  the  tension  at  212°  F.  is  1  atmosphere,  at  249°  F.  is  2  at- 
mospheres,  at  291°  F.   is   4  atmospheres,  and   at   339°  F.   is  8 
atmospheres.     Hence,  if  vapor  should  receive  its  full   increment 
of  tension,  while  the  thermometer  rises  10  degrees  from  80°  to 
90°,  a  vastly  greater  quantity  would  be  added  than  when  it  rises 
10  degrees  from  40°  to  50°.     On  the  contrary,  if  vapor  is  at  its 
full  tension  in  each  case,  much  more  water  will  be  precipitated  iu 
cooling  from  90°  to  80J  than  from  50°  to  40°. 


344  HEAT. 

546.  Dew-point. — This  is  the  temperature  at  which  vapor, 
in  a  given  case,  is  precipitated  into  water  in  some  of  its  forms. 
If  there  was  no  air,  the  dew-point  would  always  be  the  same  as 
the  existing  temperature  ;  since  lowering  the  temperature  in  the 
least  degree  would  require  a  diminished  tension  or  quantity  of 
vapor,  some  must  therefore  be  condensed  into  water.     But  in  the 
air  the  tension  may  not  be  at  its  full  height,  and  therefore  the 
temperature  may  need  to  be  reduced  several  degrees  before  pre- 
cipitation will  take  place.     A  comparison  of  the  temperature  with 
the  dew-point  is  one  of  the  methods  employed  for  measuring  the 
humidity  of  the  air. 

547.  Measure  of  .Vapor. — The  measure  of  the  vapor  exist- 
ing at  a  given  time,  is  expressed  by  two  numbers,  one  indicating 
its  tension, — i.  e.,  the  height  of  the  column  of  mercury  which  it 
will  sustain;  the  other,  humidity, — i.e.,  its  quantity  per  cent.,  as 
compared  with  the  greatest  possible  amount  at  that  temperature. 
Thus,  tension  =  0.6,  humidity  =  83,  signifies  that  the  quantity  of 
vapor  is  sufficient  to  support  six-tenths  of  an  inch  of  mercury,  and 
is  83  hundredths  of  the  quantity  which  could  exist  at  that  tem- 
perature.   The  greatest  tension  possible  at  zero  R,  is  0.04  ;  at  the 
freezing  point,  0.18  ;  at  80°  F.,  1.0.    At  the  lowest  natural  tempera- 
tures, the  maximum  tension  is  doubled  every  12°  or  14°  ;  at  the 
highest,  every  21°  or  22°. 

548.  Hygrometers.  —  This  is  the  name  usually  given  to 
instruments  intended  for  measuring  the  moisture  of  the  air.    But 
the  one  most  used  of  late  years  is  called  the  psychrometer,  which 
gives  indication  of  the  amount  of  moisture  by  the  degree  of  cold 
produced  in  evaporation ;  for  evaporation  is  more  rapid,  and  there- 
fore the  cold  occasioned  by  it  the  greater,  according  as  the  air 
is  drier.      The  psychrometer  consists  of  two  thermometers,  one 
having  its  bulb  covered  with  muslin,  which  is  kept  moistened  by 
the  capillary  action  of  a  string  dipping  in  water. 

The  wet-bulb  thermometer  will  ordinarily  indicate  a  lower 
temperature  than  the  dry-bulb ;  if,  in  a  given  case,  they  read 
alike,  the  humidity  is  100.  The  instrument  is  accompanied  by 
tables,  giving  tension  and  humidity  for  any  observation. 

Various  formulae  and  complete  tables  may  be  found  in  the 
•'Smithsonian  Meteorological  and  Physical  Tables." 

549.— Dew.— Frost.— The  deposition  called  dew  takes  place 
on  the  surface  of  bodies,  by  which  the  air  is  cooled  below  its 
dew-point.  It  is  at  first  in  the  form  of  very  small  drops,  which 
unite  and  enlarge  as  the  process  goes  on.  Dew  is  formed  in  the 
evening  or  night,  when  the  surfaces  of  bodies  exposed  to  the  sky 


FOG.  345 

become  cold  by  radiation.  As  soon  as  their  temperature  has  de- 
scended to  the  dew-point,  the  stratum  of  air  contiguous  to  them 
deposits  moisture,  and  continues  to  do  so  more  and  more  as  the 
cold  increases. 

Of  two  bodies  in  the  same  situation,  that  will  receive  most  dew 
which  radiates  most  rapidly.  Many  vegetable  leaves  are  good 
radiators,  and  receive  much  dew.  Polished  metal  is  a  poor  radia- 
tor, and  ordinarily  has  no  dew  deposited  on  it. 

Sometimes,  however,  good  radiators  have  little  dew,  because  they 
are  so  situated  as  to  obtain  heat  nearly  as  fast  as  they  radiate  it. 
Dew  is  rarely  formed  on  a  bed  of  sand,  though  it  is  a  good  radia- 
tor, because  the  upper  surface  gets  heat  by  conduction  from  the 
mass  below.  Dew  is  not  formed  on  water,  because  the  upper 
stratum  sinks  and  gives  place  to  warmer  ones. 

Bodies  most  exposed  to  the  open  sky,  other  things  being  equal, 
have  most  dew  precipitated  on  them.  This  is  owing  to  the  fact, 
that  in  such  circumstances,  they  have  no  return  of  heat  either  by 
reflection  or  radiation.  If  a  body  radiates  its  heat  to  a  building, 
a  tree,  or  a  cloud,  it  also  gets  some  in  return,  both  reflected  and 
radiated.  Hence,  little  dew  is  to  be  expected  in  a  cloudy  night, 
or  on  objects  surrounded  by  high  trees  and  buildings. 

Wind  is  unfavorable  to  the  formation  of  dew,  because  it  mingles 
the  strata,  and  prevents  the  same  mass  from  resting  long  enough 
on  the  cold  body  to  be  cooled  down  to  the  dew-point. 

When  the  radiating  body  is  cooled  below  the  freezing  point, 
the  water  deposited  takes  the  solid  form  in  fine  crystals,  and  is 
called  frost.  Frost  will  often  be  found  on  the  best  radiators,  or 
those  exposed  to  the  open  sky,  when  only  dew  is  found  elsewhere. 

550.  Fog. — This  form  of  precipitation  consists  of  very  small 
globules  of  water  sustained  in  the  lower  strata  of  the  air.  Fog 
occurs  most  frequently  over  low  grounds  and  bodies  of  water, 
where  the  humidity  is  likely  to  be  great.  If  air  thus  humid  mixes 
with  air  cooled  by  neighboring  land,  even  of  less  humidity,  there 
will  probably  be  more  vapor  than  can  exist  at  the  intermediate 
temperature,  for  the  reason  mentioned  in  Art.  545.  The  case  may 
be  illustrated  thus.  Let  two  masses  of  air  of  equal  volumes  be 
mixed,  the  temperature  of  one  being  40°  F.,  the  other  60°  F.,  and  each 
containing  vapor  at  the  highest  tension.  Then  the  mixture  will 
have  the  mean  temperature  of  50°,  and  the  vapor  of  the  mixture 
will  also  be  the  arithmetical  mean  between  that  of  the  two  masses. 
But,  according  to  the  law  (Art.  545),  the  vapor  can  only  have  a 
tension  which  is  nearly  a  geometrical  mean  between  the  two,  and 
that  is  necessarily  lower  than  the  arithmetical  mean  ;  hence  the 
excess  must  be  precipitated.  If  8  Ibs.  of  vapor  were  in  one  volume 


346  HEAT. 

and  18  Ibs.  in  the  other,  an  equal  volume  of  the  mixture  would' 
have  i  (8  4-  18)  =  13  Ibs.  of  moisture  ;  but  at  the  mean  tempera- 
ture of  50°,  only  V  8  x  18  =  12  Ibs.  could  exist  as  vapor ;  there- 
fore one  pound  must  be  precipitated.  And  even  if  one  of  the 
masses  had  a  humidity  somewhat  below  100,  still  some  precipita- 
tion is  likely  to  take  place. 

551.  Cloud. — The  same  as  fog,  except  at  a  greater  elevation. 
Air  rising  from  heated  places  on  the  earth,  and  carrying  vapor 
with  it,  is  likely  to  meet  with  masses  much  colder  than  itself,  and 
depositions  of  moisture  are  therefore  likely  to  take  place.     Moun- 
tain-tops are  often  capped  with  clouds,  when  all  around  is  clear. 
This  liappens  when  lower  and  warmer  strata  are  driven  over  them, 
and  thus  cooled  below  the  dew-point.     The  same  air,  as  it  con- 
tinues down  the  other  side,  takes  up  its  vapor  again,  and  is  as 
transparent  as  it  was  before  ascending.     A  person  on  the  summit 
perceives  a  chilly  fog  driving  by  him,  but  the  fog  was  an  invisible 
vapor  a  few  minutes  before  reaching  him,  and  returns  to  the  same 
condition  soon  after  leaving  him.     The  cloud  rests  on  the  moun- 
tain ;  but  all  the  particles  which  compose  it  are  swiftly  crossing 
over.     Clouds  are  often  above  the  limit  of  perpetual  frost ;  they 
then  consist  of  crystals  of  ice. 

552.  Rain,  Mist. — Whether  the  precipitated  moisture  has  the 
form  of  cloud  or  rain,  depends  on  the  rapidity  with  which  precipi- 
tation takes  place.     If  currents  of  air  are  in  rapid  motion,  if  the 
temperature  of  masses,  brought  into  contact  by  this  motion,  are 
widely  different,  and  if  their  humidity  is  at  a  high  point,  the  vapor 
will  be  precipitated  so  rapidly  that  the  globules  will  touch  each 
other,  and  unite  into  larger  drops,  which  cannot  be  sustained. 
Globules  of  fog  and  cloud,  however,  are  specifically  as  heavy  as, 
drops  of  rain  ;    but  they  are  sustained  by  the  slightest  upward 
movements  of  the  air,  because  they  have  a  great  surface  compared 
with  their  weight.     A  globule  whose  diameter  is  100  times  less 
than  that  of  a  drop  of  rain,  meets  with  100  times  more  obstruc- 
tion proportionally,  since  the  weight  is  diminished  a  million  times 
(T^)S>  an^  ^ne  section  only  ten  thousand  times  (T^)s.     So  the 
dust  of  even  heavy  minerals  is  sustained  in  the  air  for  some  time,, 
when  the  same  substances,  in  the  form  of  sand,  or  coarse  gravel, 
fall  instantly. 

If  a  cloud  of  fine  dust  contains  so  much  matter  as  to  make  the 
mass  of  a  cubic  foot  of  the  dusty  air  greater  than  that  of  a  cubic- 
foot  of  pure  air,  it  will  descend.  If  the  mean  density  of  a  fog 
is  greater  than  that  of  the  purer  surrounding  air,  it  will  settle 


DRAUGHT     OF    FLUES.  347 

down  into  hollows  and  valleys  ;  if  its  mean  density  is  less  than  the 
air,  it  will  rise  as  cloud. 

Mist  is  fine  rain ;  the  drops  are  barely  large  enough  to  make 
their  way  slowly  to  the  earth. 

553.  Hail,  Sleet,  Snow.— When  the  air  in  which  rapid  pre- 
cipitation occurs,  is  so  cold  as  to  freeze  the  drops,  hail  is  produced. 
As  hailstones  are  not  usually  in  the  spherical  form  when  they 
reach  the  earth,  it  is  supposed  that  they  are  continually  receiving 
irregular  accretions  in  their  descent  through  the  vapor  of  the  air. 
Hail-storms  are  most  frequent  and  violent  in  those  regions  where 
hot  and  cold  bodies  of  air  are  most  easily  mixed.     Such  mixtures 
are  rarely  formed  in  the  torrid  zone,  since  there  the  cold  air  is  at  a 
great  elevation ;  in  the  frigid  zone,  no  hot  air  exists  at  anv  height ; 
but  in  the  temperate  climates,  the  heated  air  of  the  torrid,  and  the 
intensely  cold  winds  of  the  frigid  zone,  may  be  much  more  easily 
brought  together;  and  accordingly,  in  the  temperate  zones  it  is 
that  hail-storms  chiefly  occur.     Even  in  these  climates,  they  are 
not  frequent  except  on  plains  and  valleys  contiguous  to  mountains 
which  are  covered  with  snow  during  the  summer.     The  slopes  of 
the  mountain  sides  give  direction  to  currents  of  air,  so  that  masses 
of  different  temperature  are  readily  mingled  together. 

Sleet  is  frozen  mist,  that  is,  it  consists  of  very  small  hailstones. 

Snow  consists  of  the  small  crystals  of  frozen  cloud,  united  in 
flakes.  Like  all  transparent  substances,  when  in  a  pulverized 
state,  it  owes  its  whiteness  to  innumerable  reflecting  surfaces.  A 
cloud,  when  the  sun  shines  upon  it,  is  for  the  same  reason  in- 
tensely white  (Art.  371). 

554.  Draught  of  Flues. — The  effect  of  the  sun's  heat  in 
causing  circulat/on  of  the  air  has  been  already  considered  (Art. 
268-272).     Similar  movements  on  a  limited  scale  are  produced 
whenever  a  portion  of  the  air  is  heated  by  artificial  means.     Thus, 
the  air  of  a  chimney  is  made  lighter  by  a  fire  beneath  it,  than  a 
column  of  the  outer  air  extending  to  the  same  height.     It  is  there- 
fore pressed  upward  by  the  heavier  external  air,  which  descends 
and  moves  toward  the  place  of  heat.     The  difference  of  weight  in 
the  two  columns  is  greater,  and  therefore  the  draught  stronger, 
if  the  chimney  is  high,  provided  the  supply  of  heat  is  sufficient 
to  maintain  the  requisite  temperature.     Chimneys  are  frequently 
built  one  or  two  hundred  feet  high  for  the  uses  of  manufactories. 
The   high  fireplaces  and   large   flues  of   former  times  were  un- 
favorable for  draught,  both  because  much  cold  air  could  mingle 
with  that  which  was  heated,   and   because   there  was  room   for 
external  air  to  descend  by  the  side  of  the  ascending  column.     For 


348  HEAT. 

good  draught,  no  air  should  be  allowed  to  enter  the  flue  except 
that  which  has  passed  through  the  fire. 

555.  Ventilation  of  Apartments.  —  The  air  of  an  apart- 
ment, as  it  becomes  vitiated  by  respiration,  may  generally  be 
removed,  and  fresh  air  substituted,  by  taking  advantage  of  the 
same  inequality  of  weight  in  air-columns,  which  has  been  men- 
tioned. If  opportunity  is  given  for  the  warm  impure  air  to  escape 
from  the  top  of  a  room,  and  for  external  air  to  take  its  place,  there 
will  be  a  constant  movement  through  the  room,  as  in  the  flue  of  a 
chimney,  though  at  a  slower  rate.  If  the  external  air  is  cold, 
the  weight  of  the  columns  differs  more,  and  therefore  the  ventila- 
tion is  more  easily  effected.  But  in  cold  weather,  the  air,  before 
being  admitted  to  the  room,  is  warmed  by  passing  through  the 
air-chambers  of  a  furnace.  When  there  is  a  chimney-flue  in 
the  wall  of  a  room,  with  a  current  of  hot  air  ascending  in  it,  the 
ventilation  is  best  accomplished  by  admitting  the  air  into  the 
flue  at  the  upper  part  of  the  room  ;  since  it  will  then  be  removed 
with  the  velocity  of  the  hot-air  current. 

The  tendency  of  the  air  of  a  warm  room  to  pass  out  near  the 
top,  while  a  new  supply  enters  at  the  lower  part,  is  shown  by 
holding  the  flame  of  a  candle  at  the  top,  and  then  at  the  bottom, 
of  a  door  which  is  opened  a  little  distance.  The  flame  bends  out- 
ward at  the  top  and  inward  at  the  bottom. 

The  impure  air  of  a  large  audience-room  is  sometimes  removed 
by  a  mechanical  contrivance,  as,  for  instance,  a  fan-wheel  placed 
above  an  opening  at  the  top,  and  driven  by  steam. 

The  ventilation  of  mines  is  accomplished  sometimes  by  a  fire 
built  under  a  shaft,  fresh  air  being  supplied  by  another  shaft,  and 
sometimes  by  a  fan-  pia  ^ 

wheel  at  the  top  of 
the  shaft.  If  there 
happen  to  be  two 
shafts  which  open  to 
the  surface  at  very 
different  elevations, 
ventilation  may  be 
effected  by  the  in- 
equality of  tempera- 
ture which  is  likely 
to  exist  within  the 
earth  and  above  it. 
Let  M  M  (Fig.  312) 
be  the  vertical  section  of  a  mine  through  two  shafts  A  and  B, 
which  open  at  different  heights  to  the  surface  of  the  earth.  If  the 


SOURCES     OF     HEAT.  349 

external  air  is  of  the  same  temperature  as  the  air  within  the  earth, 
then  the  column  A  in  the  longer  shaft  has  the  same  weight  as  B 
and  C  together,  measured  upward  to  the  same  level.  In  that  case, 
which  is  likely  to  occur  in  spring  and  fall,  there  is  no  circulation 
without  the  use  of  other  means.  But  in  summer  the  air  C  is 
warmer  than  A  and  B ;  therefore  A  is  heavier  than  B  +  C.  Hence 
there  is  a  current  of  air  down  A  and  up  B.  In  winter,  C  is  colder 
than  air  within  the  earth ;  therefore  B  +  C  are  together  heavier 
than  A,  and  the  current  sets  in  the  opposite  direction,  down  B 
and  up  A. 


556.  Sources  of  Heat.— The  sun,  although  nearly  a  hun- 
dred millions  of  miles  from  the  earth,  is  the  source  of  nearly  all 
the  heat  existing  at  its  surface.  The  interior  of  the  earth,  except 
n  thickness  of  forty  or  fifty  miles  next  to  the  surface,  is  believed  to 
be  in  a  condition  of  heat  so  intense  that  all  the  materials  compos- 
ing it  are  in  the  melted  state.  But  the  earth's  crust  is  so  poor  a 
conductor  that  only  an  insensible  fraction  of  all  this  heat  reaches 
the  surface. 

The  energy  radiated  to  the  earth  by  the  sun  amounts  to  83 
foot-pounds  per  square  foot  of  the  earth's  surface  per  second.  Sir 
William  Thomson,  in  his  memoir  on  the  "  Mechanical  Value  of  a 
Cubic  Mile  of  Sunlight,"  says  that  the  energy  of  the  waves  com- 
prised within  a  cubic  mile  of  ether  near  the  earth  is  equal  to  about 
12,050  foot-pounds. 

Mechanical  operations  are  always  attended  by  a  development 
of  heat.  For  example,  if  a  broad  surface  of  iron  were  made  to 
revolve,  rubbing  against  another  surface,  nearly  all  the  energy  ex- 
pended in  overcoming  the  friction  would  appear  as  heat,  a  com- 
parative!}* small  part  being  conveyed  through  the  air  as  sound. 
The  cutting  tool  employed  in  turning  an  iron  shaft  has  been 
known  to  generate  heat  enough  to  raise  a  large  quantity  of  cold 
water  to  the  boiling-point,  and  to  keep  it  boiling  for  an  indefinite 
time.  It  is  a  fact  familiar  to  all,  that  violent  friction  of  bodies 
against  each  other  will  set  combustibles  on  fire.  The  axles  of 
railroad  cars  are  made  red-hot  if  not  duly  oiled  ;  boats  are  set 
on  fire  by  the  rope  drawn  swiftly  over  the  edge  by  a  whale  after 
he  is  harpooned  ;  a  stream  of  sparks  flies  from  the  emery  wheel 
when  steel  is  polished,  etc. 

A  lecture  illustration  devised  by  Tyndall  will  show  the  con- 
version of  motion  into  heat. 

Screw  a  brass  tube  A,  about  4  inches  long  and  •£  inch  in  diam- 
eter, upon  the  spindle  of  a  whirling  table  B  (Fig.  313).  Nearly 


350  HEAT. 

fill  the  tube  with  water  and  insert  a  cork ;  press  the  tube  be- 
tween the  jaws  of  a  wooden  clamp  C,  while  A  is  rapidly  rotating. 
Heat  is  developed  by  the  FlG  313 

friction,  and  this  communi- 
cated  to  the  water  causes 
it  to  boil,  and  finally  to  eject 
the  cork. 

The  heat  developed  by 
sudden  compression  of  air 
may  be  rendered  visible 
by  igniting  vapor  of  car- 
bon bisulphide  in  a  "Fire 
Syringe."  A  thick  glass  tube  A  (Fig.  314)  closed  at  the  lower 
end,  has  a  well-fitted  piston  C,  whose  rod  is  terminated  by  a  wide 
cap,  or  button,  B,  upon  which  the  palm  of  the  hand  may 
FIG.  314.  strike  forcibly  without  injury.  If  a  bit  of  tinder,  or  a 
small  tuft  of  cotton  moistened  with  carbon  bisulphide, 
be  placed  at  the  bottom  of  the  tube,  it  will  be  ignited 
when  the  piston  is  driven  down  by  a  sudden  blow 
upon  it. 

Chemical  action  is  another  very  common  source  of 
heat.  Combustion  is  the  effect  of  violent  chemical  at- 
traction between  atoms  of  different  natures,  when  both 
light  and  heat  are  manifested.  If  the  union  goes  on 
slowly,  as  in  the  rusting  of  iron,  the  amount  of  heat 
is  the  same,  but  it  is  diffused  as  fast  as  developed.  The 
molecular  energy,  in  most  cases  of  chemical  combina- 
tion, as  measured  by  the  heating  effect,  is  enormously 
great. 

The  warmth  produced  by  the  vital  processes  in  plants 
and  animals  is  caused  by  chemical  action.  In  breath- 
ing the  air,  some  of  its  oxygen  is  consumed,  which  be- 
comes united  with  the  blood.  This  process  is  in  some  respects 
analogous  to  a  slow  combustion,  by  which  heat  is  evolved  in  the 
animal  system. 


PART   VII. 

ELECTRICITY    AND    MAGNETISM. 


CHAPTER    I. 

ELECTROSTATICS.— POTENTIAL.— CAPACITY. 

557.  Definition. — The  name  Electricity,  from  the  Greek  word 
for  amber,  is  given  to  a  peculiar  agency,  which  causes  mutual 
attractions  or  repulsions  between  light  bodies,  and  which,  under 
proper  conditions,  also  produces  heat,  light,  sound,  and  chemical 
decomposition. 

Lightning  and  thunder  are  familiar  illustrations  of  the  intense 
action  of  this  agency. 

558.  Common  Indications  of  Electricity. — If  amber,  seal- 
ing-wax, or  any  other  resinous  substance,   be  rubbed  with  dry 
woollen  cloth,  fur,  or  silk,  and  then  brought  near  the  face,  the 
excited  electricity  disturbs  the  downy  hairs  upon  the  skin,  and 
thus  causes  a  sensation  like  that  produced  by  a  cobweb.     When 
the  tube  is  strongly  excited,  it  gives  off  a  spark  to  the  finger  held 
toward  it,  accompanied  by  a  sharp  snapping  noise.     A  sheet  of 
writing-paper,  first  dried  by  the  fire,  and  then  laid  on  a  table 
and  rubbed  with  India-rubber,  becomes  so  -much  excited  as  to 
adhere  to  the  wall  of  the  room  or  any  other  surface  to  which  it  is 
applied.     As  the  paper  is  pulled  up  slowly  from  the  table  by  one 
edge,  a  number  of  small  sparks  may  be  seen  and  heard  on  the 
under  side  of  the  paper.     In  dry  weather,  the  brushing  of  a  gar- 
ment causes  the  floating  dust  to  fly  back  and  cling  to  it 

Bodies  are  said  to  be  electrically  excited  when  they  show  signs 
of  electricity  in  consequence  of  some  mechanical  action  performed 
upon  them,  as  in  the  experiments  already  described. 

A  body  is  electrified  when  it  receives  electricity,  by  communica- 
tion, from  another  body  already  excited  or  electrified. 

559.  Repulsion.  —  An   electrically   excited    body   does    not 
always  produce  attraction.     It  will  be  noticed  that  pith-balls,  after 


352  ELECTRICITY     AND     MAGNETISM. 

coming  in  contact  with  an  electrified  body,  which  has  attracted 
them,  are  repelled.  They  have  received  a  portion  of  the  elec- 
tricity which  attracted  them  and  repulsion  is  the  result.  This 
repulsion  can  be  made  much  more  apparent  if  an  electrified  vul- 
canite rod  be  suspended  in  a  wire  loop  at  the  end  of  a  silk  thread 
and  then  a  similarly  electrified  rod  be  approached  to  it.  The 
suspended  rod  can  be  made  to  revolve  rapidly  because  of  the 
repulsion. 

560.  Theories  of  Electricity.— As  to  the  exact  nature  of 
electricity  science  is  still  in  the  dark,  though  probably  the  dark- 
ness which  precedes  dawn. 

Symmer  proposed  a  "two-fluid"  theory.  He  supposed  every 
unelectrified  body  to  contain  equal  quantities  of  two  opposite 
kinds  of  imponderable  electric  fluid.  In  equal  quantities  they 
neutralized  each  other.  But  if,  by  friction  or  other  means,  the 
amount  of  one  fluid  be  made  to  exceed  that  of  the  other,  then 
the  body  becomes  positively  or  negatively  electrified.  According 
to  this  theory  two  positively  or  two  negatively  electrified  bodies 
repel  each  other ;  a  positively  electrified  body  and  a  negatively 
electrified  body  attract  each  other. 

Franklin  modified  this  into  a  "  one-fluid  "  theory.  Every  body 
contains  its  own  normal  amount  of  one  electric  fluid.  This  amount 
is  increased  or  decreased  when  rubbed  by  another  body.  The 
surplus  amourut  is  obtained  from  or  given  up  to  this  second  body. 
The  body  with  more  than  its  normal  amount  is  positively  elec- 
trified, and  negatively  electrified  when  it  has  less  than  this  amount. 

Lodge  maintains  that  electricity  is  the  luminiferous  ether  itself. 
He  arrives  at  this  conclusion  after  cpnsidering  a  great  number  of 
electrical  phenomena  which  demand  the  ether  for  their  proper 
explanation.- 

Without  adopting  any  theory,  electrical  laws  and  phenomena 
may  be  understood  by  considering  the  fact  that  a  body  may  be 
subject  to  two  opposite  electrical  conditions.  It  may  be  positively 
or  negatively  electrified.  The  law  regarding  attraction  and  repul- 
sion then  is  : 

Similarly  electrified  bodies  repel  each  other,  and  dissimilarly  elec- 
trified bodies  attract  each  other. 

561.  Electric   Series. — If  two  bodies  are  rubbed  together, 
one  of  them  is  electrified  positively  and  the  other  negatively.     One 
of  these  bodies,  if  rubbed  by  a  third,  may  be  oppositely  electrified 
to  what  it  was  in  the  first  case.     Silk,  when  rubbed  with  glass,  is 
negatively  electrified  ;  but  rubbed  with  sulphur,  it  receives  a  pos- 
itive  charge.      In   the    following    series   each   member1  becomes 


COULOMB'S    LAW. 


358 


positively  charged  when  rubbed  on  one  following  it,  negatively 
when  rubbed  on  one  preceding  it :  fur,  wool,  rezin,  glass,  cotton, 
silk,  wood,  metals,  sulphur,  india-rubber,  gutta-percha. 

562.  Conductors  and  Insulators. — When  a  glass  or  vul- 
canite tube  is  rubbed  with  cat's  fur,  it  shows  that  it  has  become 
electrified  by  attracting  light  articles.  If  a  metal  rod  be  substi- 
tuted for  the  glass  one,  no  attraction  will  be  evidenced.  This  is 
not  because  the  metal  was  not  electrified  by  the  rubbing,  but 
because  the  electricity,  as  soon  as  generated,  escaped,  through  the 
rod  itself  and  the  hand  holding  it,  to  the  ground.  If  the  rod  be 
held  by  a  glass  or  hard-rubber  handle  and  then  rubbed,  it  will 
attract  as  the  glass  did.  This  shows  that  some  substances,  as 
metals,  allow  electricity  to  pass  freely  through  them,  while  others, 
as  glass,  almost  entirely  prevent  its  passage.  The  first  class  of 
substances  are  ^called  conductors,  the  latter  class  non-conductors  or 
insulators.  Some  substances  neither  conduct  nor  insulate  well,  but 
lie  between  the  two  classes.  The  following  is  a  table  of  substances 
arranged  in  the  order  of  their  electrical  conductivity  : 


CONDUCTORS.  |  4.  Acids. 

1.  Metals.  !  5.  Sea-water. 

2.  Charcoal.  |  6.  Vegetables. 

3.  Graphite.  |  7.  Animals. 


8.  Wood. 

9.  Silk. 

10.  India-ruhber. 

11.  Porcelain. 


12.  Glass. 

13.  Shellac. 

14.  Vulcanite. 
INSULATORS. 


A  conductor  mounted  upon  or  suspended  by  an  insulator  is  said 
to  be  insulated. 

A  method  for  determining  the  conductivity  of  substances  is  to 
suspend  two  pith-balls  by  moistened  threads  from  a  metal  insulated 
hook.  Upon  communicating  a  charge  of  electricity  to  the  balls 
they  will  stand  out  away  from  each  other,  owing  to  the  repulsion 
between  the  same  kinds  of  electricity  on  each.  If,  now,  one  end  of 
the  substance,  whose  conductivity  is  to  be  determined,  be  held  in 
the  hand  and  the  other  be  touched  to  the  hook  from  which  the 
balls  are  suspended,  the  rapidity  with  which  the  balls  fall  toward 
each  other  determines  the  conductivity.  If  they  fall  instantly,  the 
substance  is  a  good  conductor.  If  they  remain  separated,  the  sub- 
stance is  a  good  insulator.  After  an  insulator  or  an  insulated  con- 
ductor has  been  charged  with  electricity,  the  electricity  of  necessity 
remains  at  rest,  and  is,  for  this  reason,  called  statical  electricity.  If, 
now,  it  be  connected,  by  means  of  a  conducting  wire,  with  the 
moist  earth,  it  will  pass  off  instantly  to  the  earth.  During  the 
time  of  its  passage  it  is  called  dynamical  electricity.  If  by  chemical 
or  other  means  the  flow  be  maintained,  then  the  dynamical  elec- 
tricity is  called  galvanic  or  voltaic. 

563.  Coulomb's  Law. — Coulomb  showed  that,  correspond- 


"354:  ELECTRICITY     AND     MAGNETISM. 

ing  to  Newton's  law  of  gravitation,  the  force  of  attraction  betiveen 
dissimilarly  electrified  bodies  and  the  force  of  repulsion  between  sim- 
ilarly electrified  bodies  is-  directly  proportional  to  the  product  of  the 
quantities  of  electricity  and  inversely  proportional  to  the  square  of  the 
distance  betiveen  the  bodies. 

If  we  represent  the  force  by  f  dynes,  the  distance  by  r  centi- 
metres, and  the  quantities  of  electricity  by  q  and  q',  then  we  can 
indicate  the  law  by  the  equation 

f  -  q  q'- 
J  ~  -75- 

If  these  magnitudes  be  connected  by  the  sign  of  equality,  a  proper 
unit  of  quantity  must  be  had.  Letting  /  =  1  dyne,  r  =  1  centi- 
meter, and  q  =  q,  then  q'  =  1  and  q  =  ^  1.  Hence  we  may  define 
the  unit  of  electrical  quantity  as  follows  : 

One  unit  of  electricity  is  that  quantity  which,  when  placed  at  a  dis- 
tance of  one  centimetre  from  a  similar  and  equal  quantity,  repels  it 
with  a  force  of  one  dyne. 

If  the  quantity  of  electricity  be  spread  over  a  body  of  some 
size,  as  a  sphere,  then  the  distance  r  must  be  measured  from  some 
point  as  the  centre  of  the  sphere.  This  is  evidently  for  the  same 
reason  as  in  gravitation,  where  the  distance  is  measured  from  the 
centre  of  gravity. 

It  must  be  borne  in  mind  that  the  unit  of  quantity  here  given 
is  based  upon  the  force  exerted  by  two  statical  quantities  of  elec- 
tricity. Another  unit,  based  upon  the  electro-magnetic  force,  will 
be  mentioned  later. 


564.  Potential.— Whenever  a  body  is  lifted  vertically  away 
from  the  eaiih,  the  work  performed  in  lifting  it  has  been  trans- 
formed into  potential  energy.  The  body  has,  because  of  the 
attraction  between  it  and  the  earth,  a  potential  energy  capable  of 
doing  exactly  the  same  number  of  ergs  or  foot-pounds  of  work  as 
were  used  in  raising  it  to  its  position  (Art.  36).  Similarly,  if  two 
conductors,  charged  with  the  same  kind  of  electricit}',  be  ap- 
proached towards  each  other,  a  certain  number  of  ergs  of  work 
will  have  been  performed,  owing  to  the  repulsion  between  them. 
(A  more  perfect  analogy  would  be  to  suppose  two  dissimilarly 
charged  conductors  to  be  separated.)  The  work  which  has  been 
performed  is  also  changed  into  potential  energy  between  the  con- 
ductors. The  amount  of  energy  made  potential  depends  upon  the 
quantities  of  electricity  on  each  of  the  conductors,  and  upon  the 
distance  through  which  they  have  been  moved  toward  each  other. 
For  energy  is  measured  by  the  work  it  can  do,  and  work  in  ergs 
equals  the  product  of  the  force  in  dynes  by  the  distance  in  centi- 


EQUIPOTENTIAL     SURFACES.  355 

metres  through  which  it  has  acted.  Now  the  force  of  repulsion 
between  the  two  conductors  equals  the  product  of  their  quantities 
divided  by  the  square  of  their  distance  apart. 

Suppose  one  of  the  conductors  to  Have  any  charge  and  to 
be  fixed  immovably.  Then  let  three  charges  of  respectively  1,  2, 
and  3  units  of  quantity  be  successively  approached,  between  the 
same  limits,  towards  the  first  conductor.  In  the  first  case  a  cer- 
tain amount  of  energy  will  have  been  made  potential  ;  in  the 
second  case  twice  as  much,  and  in  the  third  three  times  as  much. 
Evidently  a  certain  amount  of  the  energy  made  potential  is  owing 
to  the  ^immovable  charge,  and  this  amount  is  the  same  in  each 
case.  This  condition  of  the  space  around  an  electrified  body  is 
termed  the  potential,  owing  to  that  charge.  To  obtain  a  quanti- 
tative expression  for  it,  the  movable  charge  must  be  taken  of  unit 
quantity.  It  must  also  be  considered  that  the  work  necessary  to 
approach  an  unit  through  a  given  distance  is  not  as  great  as  to 
approach  it  through  twice  that  distance.  Considering  these  two 
points,  we  have  the  definition  of  electrostatic  potential  : 

The  potential  at  any  point  is  the  work  that  must  be  spent  upon  a 
unit  of  positive  electricity  in  bringing  it  up  to  that  point  from  an 
infinite  distance. 

If  the  immovable  charge  be  negative,  no  work  would  be 
required  to  move  up  a  positive  unit  ;  on  the  contrary,  work  would 
be  performed  by  the  unit  in  travelling.  Hence  the  potential, 
owing  to  a  negative  charge,  is  negative  potential.  It  is  convenient 
to  consider  it  so. 


565.  Equipotential  Surfaces.  —  If  the  charge  of  electricity 
be  supposed  to  lie  on  a  small  sphere,  then  some  point  can  be 
found  on  every  possible  radius  of  the  sphere  produced  where  the 
potential  will  be  the  same.  That  is,  it  would  require  the  same 
amount  of  work  to  bring  a  positive  unit  of  electricity  from  an 
infinite  distance  out  on  each  radius  to  this  point.  In  the  case  of  a 
sphere  being  charged,  these  points  would  be  equally  distanced 
from  the  centre  of  the  sphere.  If  now  these  points  be  connected 
together,  a  spherical  surface  will  result.  Any  such  surface  which 
contains  onlv  points  of  the  same  potential  is  called  an  equipotential 
surface. 

In  order  that  an  equipotential  surface  may  be  spherical,  the 
charge  must  lie  upon  a  sphere  and  must  be  free  from  other  electri- 
fied bodies.  If  the  electrified  body  be  irregular  in  shape,  the 
equipotential  surfaces  will  be  correspondingly  so. 

To  transfer  a  quantity  of  electricity  from  one,  point  in  an  equi- 
potential surface  to  another  in  the  same  surface  requires,  no  work  to 


356  ELECTRICITY     AND     MAGNETISM. 

be  performed.  For  while  it  may  require,  work  to  move  the  charge- 
in  one  direction  from  the  surface,  it  will  require  a  negative  expen- 
diture of  work  to  bring  it  back  again,  i.e.,  the  attraction  or  repul- 
sion between  the  electricities  performs  the  work. 

566.  Difference  of  Potential. — In  the  consideration  of  most 
problems  in  electricity  involving  the  idea  of  potential,  the  potential 
of  two  points  is  required.     However,  it  is  not  the  absolute  poten- 
tial of  each  of  the  points,  but  the  difference  of  potential  between 
them  which  is  considered.     If  it  requires  a  certain  number  of  ergs 
to  bring  a  unit  of  positive  electricity  from  an  infinite  distance  up 
to  a  given  point,  and  more  ergs  to  bring  it  up  to  a  second  point, 
then  this  extra  work  is  what  would  be  required  to  move  the  unit 
from  the  first  to  the  second  point     This  number  of  ergs  is  then 
the  measure  of  the  difference  of  potential  between  the  two  points. 
Hence  we  obtain  the  definition  : 

The  unit  difference  of  potential  is  that  which  must  exist  between 
two  points,  that  one  erg  may  be  required  to  move  a  positive  unit 
of  electricity  from  one  to  the  other. 

567.  Unit  of  Potential. — The  difference  of  potential  between 
two  points,  a  and  b,  Fig.  315,  at  distances  r  and  r'  from  a  quantity 


FIG.  315. 


of  electricity  q,  is  measured  by  the  work  necessary  to  move  a 
positive  unit  of  electricity  from  b  to  a. 

This 

work  =  (average)  force  x  distance  through  which  it  is  overcome. 
The  distance  =  r'  —  r. 


Force  at  a  =  ~ 

r- 

Force  at  6  =  -4r 


/    <?'  Q* 

average  force  =  -y    ^   n  —  -?—, 

Hence  the  difference  of  potential 


This  equation  for  the  difference  of  potential  between  two  points 
enables  us  to  obtain  an  equation  for  the  absolute  potential  Va  at 

*  That  this  is  a  true  average  can  be  proved  by  a  simple  application  of  the 
calculus. 


CAPACITY.  ;•{.-,  7 

any  point  a.  We  have  only  to  suppose  that  the  second  point  b  is 
removed  to  an  infinite  distance,  where  its  potential  Vb  =  0,  and 
r'  =.  oo  .  Hence 

1  "•  =  f 

Or,  in  general, 

The  potential,  F,  of  any  point  at  a  distance,  r,  from  a  quantity  of 
electricity,  q,  is  expressed  by  the  equation, 


From  this  equation,  supposing  q  and  r  each  equal  to  unity,  we  ob- 
tain the  definition  : 

The  unit  potential  is  thai  dur  to  a  unit  quantity  of  electricity  at  a 
distance  of  one  centimetre. 

The  potential  at  a  point  owing  to  several  charges  of  electricity 
is  equal  to  the  sum  of  the  potentials  at  that  point  due  to  each 
charge  taken  separately.  Thus,  if  quantities  of  electricity  <?,  q',  and 
q"  a  -e  at  distances  ?*,  r',  and  r"  from  a  point,  the  potential  at  that  point 

' 


568.  Zero  Potential.  —  At  an  infinite  distance  away  from  any- 
electrified  body  the  potential  would  evidently  be  zero.     If  a  pos- 
itive charge  were  brought  near,  the  potential  would  become  positive, 
and  negative  for  a  negative  charge.     In  practice  it  is  convenient  to 
take  the  earth  as  a  standard  zero,  with  which  all  other  potentials 
may  be  compared.     This  assumption  is  analogous  to  the  use  of  the 
sea-level  as  the  zero  in  measuring  the  heights  of  mountains  instead 
of  the  centre  of  the  earth, 

569.  Potential   on    a    Sphere.  —  By   discussing    the  equa- 
tions in  Art.  567,  and  supposing  r  equal  to  zero,  one  might  be  led 
to  think  that  the  potential  would  be  infinite.     But  it  must  be 
remembered  that,  just  as  in  gravitation,  there  is  a  centre  from 
which  all  electric  force  apparently  works.     In  the  case  of  the  earth 
all  attraction  is  toward  the  centre  of  gravity.     With  an  electrified 
sphere  all  action  cornes  from  the  centre  of  the  sphere.     The  elec- 
tricity (Art.  572)  lies  upon  the  surface  of  it,  but  the  resultant  of 
all  attractions  from  all  the  particles  of  electricity  passes  through 
the  centre.     Thus  a  point  in  the  electricity  itself  on  a  charged 
conductor  has  a  finite  potential,  and,  in  the  case  of  a  sphere,  it  IK 
equal  to  the  quantity  of  electricity  divided  by  the  radius  of  the  sphere. 

570.  Capacity.  —  Suppose  that  a  point  on   the  surface  of  a 
charged  spherical'  conductor,  i.e.,  any  point  in  the  electricity  itself, 


358  ELECTRICITY    AND     MAGNETISM. 

to  have  a  certain  potential.  If,  now,  the  radius  of  the  sphere  be 
supposed  to  grow  smaller,  while  the  quantity  of  electricity  remains 
the  same,  then  the  potential  will  evidently  increase  as  the  radius 

decreases,  because  the  potential  V  —  —  •     Thus  a  large  sphere,  e.g., 

the  earth,  can  hold  a  large  quantity  of  electricity  without  having  a 
high  potential.  This  ratio  between  quantity  and  potential  of  elec- 
tricity in  a  conductor  is  termed  the  electrostatic  capacity  of  the 
conductor.  Representing  this  by  C,  we  have  the  definition  in  the 
form  of  an  equation  : 

<7=£ 
V 

From  this,  by  supposing  Q  and  V  each  equal  to  unity,  we  have  the 
definition, 

That  conductor  has  a  unit  of  electrostatic  capacity,  which  requires 
a  unit  quantity  of  electricity  to  raise  its  potential  from  zero  to  one. 

Applying  the  above  equation  to  a  sphere  of  radius  r  we  have 

r=£=<i 

C         r 

whence  we  see  that  the  electrostatic  capacities  of  spheres  are 
equal  to  their  radii.  Accordingly  a  sphere  of  1  centimetre  radius 
has  a  unit  capacity. 

A  conductor,  no  matter  what  its  shape,  will  have  a  capacity, 
and  we  may  say  that, 

The  capacity  of  any  conductor  is  equal  to  the  number  of  units 
of  quantity  of  electricity  necessary  to  raise  its  potential  from  zero  to 
unity. 

571.  Equipotential  of  Connected  Conductors. — When  a 
conductor  is  charged  with  electricity  each  particle  strives  to  get 
out  of  the  reach  of  its  neighbors,  because  of  the  natural  repulsion 
between  like  kinds  of  electricity.  The  particle,  however,  cannot 
escape,  because  the  dry  air  is  an  insulator.  If,  now,  it  be  con- 
nected, by  means  of  a  conducting  wire,  with  the  earth,  the  par- 
ticle, followed  by  others,  will  flow  off  to  the  earth.  Now  the  earth, 
having  such  a  very  large  radius,  would  require  an  enormous 
quantity  of  electricity  to  raise  its  potential  even  an  infinitesimal 
amount.  The  result  is  that  the  potential  of  the  conductor  and 
earth  are  both  reduced  to  zero.  Suppose,  however,  that  instead 
of  being  connected  with  the  earth  it  had  been  connected  to  another 
insulated  conductor.  The  particles  escaping  from  the  first  con- 
ductor would  gradually  raise  the  potential  of  the  second  until  a 
particle,  at  some  place  on  the  connecting  wire,  would  be  equally 
repelled  by  the  charges  on  each  conductor,  and  would  accordingly 


DISTRIBUTION.  359 

remain  at  rest.  Now  it  will  be  found  that,  just  as  when  two 
vessels,  one  of  which  contains  water,  when  connected  by  a  tube  at 
the  bottom,  will  allow  the  flow  of  water  until  the  level  in  -both  is 
the  same,  so  with  these  conductors,  the  potentials  of  both  will  have 
become  the  same  because  of  the  connecting  wire.  Furthermore, 
just  as  is  the  case  with  the  connected  vessels  of  water,  it  makes  no 
difference  whether  the  second  conductor  had  originally  a  charge  of 
electricity  or  not.  The  potential  of  all  electrostatically  charged 
conductors  becomes  the  same  when  connected  together. 

If  the  potential  of  connected  conductors  becomes  the  same, 
then  it  is  quite  evident  the  total  quantity  of  electricity  must  be  so 
divided  that  each  conductor  shall  have  a  quantity  in  direct  propor- 
tion to  its  capacity.  This  must  necessai'ily  follow  from  the  defi- 
nition of  capacity  at  the  end  of  Art.  570.  Thus  three  connected 
conductors  of  capacities  1,  2,  and  3  would  have  respectively  one-, 
two-,  and  three-sixths  of  the  total  quantity  of  electricity  upon  them. 

572.  Position  of  Static  Charge. — A  statical  charge  of  elec- 
tricity always  resides  on  the  outside  surface  of  a  conductor.     It 
also  resides  on  the  outside  of  the  geometrical  figure  of  the  con- 
ductor.    Thus,  if  a  charge  be  communicated  to  a  wire  bird-cage, 
it  will  reside  wholly  on  the  outside  half  of  the  wires  and  none  will 
lie  on  the  inside.     This  may  be  shown  in  many  ways. 

If  a  hollow,  insulated,  conducting  cylinder  (Fig.  316)  be  pro- 
vided with  two  suspended  pith  balls  in  the  interior  and  two  on  the 
exterior,  and  a  charge  of  electricity  be  communi- 
cated to  it,  the  outside  balls  will  diverge,  owing 
to  the  repulsion  of  like  electricities.  The  inside 
balls  will,  on  the  other  hand,  remain  at  rest.  It 
makes  no  difference  whether  the  charge  be  com- 
municated to  the  inside  or  outside.  As  soon  as  it 
has  been  communicated  the  inside  balls  drop  to 
their  normal  position. 

In  calculating  the  capacities  of  conductors,  it 
makes  no  difference  whether  a  conductor  is  solid 
or  hollow. 

573.  Distribution   of    a   Charge   on   the 
Surface. — Statical  electricity  resides  at  the  sur- 
face of  a  body,  as  we  have  seen,  but  is  not  uni- 
formly diffused  over  it,  except  in  the  case  of  the  sphere.     In  gen- 
eral, the  more  prominent,  the  part,  and  the  more  rapid  its  curvature, 
the  more  intensely  is  the  electricity  accumulated  there. 

In  a  long,  slender  rod  the  density  is  greatest  at  the  ends,  nearly 
the  whole  charge  being  collected  at  these  points.  On  a  sphere, 


360  ELECTRICITY     AND     MAGNETISM. 

not  influenced  by  other  electrified  bodies,  the  density  is  uniform, 
as  illustrated  in  Fig.  317,  the  dotted  line  denoting  by  its  constant 
distance  from  the  surface  the  uniform  distribution  of  the  charge. 
Fig.   318  represents  the  varying   density  upon   an   ellipsoid 

FIG.  317.  FIG.  318. 


The  two  ellipsoids  are  similar,  and  the  ellipsoidal  shell  included 
between  them  represents  the  densities  at  various  points.  In  this 
case  the  densities  at  any  two  points  of  the  ellipsoid  are  nearly 
proportional  to  the  diameters  through  those  points. 

The  student  must  remember  that  the  charge  does  not  form  a 
layer  upon  the  body  in  any  sense  whatever,  and  that  the  above 
figures  are  given  merely  to  aid  the  memory  in  retaining  the  law 
of  distribution. 

574.  Surface  Density.  —  The  greater  the  quantity  of  elec- 
tricity on  a  given  conductor,  the  greater  the  tendency  is  for  the 
electricity  to  escape  to  surrounding  objects. 

The  surface  density  at  any  point  of  a  surface,  when  the  dis- 
tribution is  uniform,  is  the  quantity  of  electricity  per  square 
centimetre  of  surface. 

If  Q  units  of  electricity  reside  on  S  square  centimetres  of  sur- 
face, then  the  surface  density  d  is  represented  by  the  formula 


The  surface  of  a  fine  point  is  very  small,  hence,  if  there  is  any 
quantity  of  electricity  supplied  to  it,  the  density  becomes  very 
great  and  the  charge  escapes  into  the  air. 

575.  Quadrant  Electrometer.  —  This  instrument  is  used  for 
determining  very  accurately  differences  of  electrical  potential.  A 
simple  form,  suited  for  qualitative  work,  is  shown  in  Fig.  319. 
Four  like  pieces  of  metal  are  suspended,  by  conducting  rods,  from 
the  insulating  top  of  a  glass  case.  They  are  symmetrically  placed 
(as  shown  in  the  small  figure)  and  are  fixed  in  position.  Over 
these  quadrants,  as  they  are  termed,  swings  a  flat,  aluminum 
needle,  suspended  by  a  wire  of  small  diameter.  This  prolonged 
suspension  hangs  in  a  glass  chimney,  placed  upon  the  top  of  the 


QUADRANT     ELECTROMETER. 


361 


case.      A  small  mirror,   J/,  is  attached   to  a  rigid  prolongation 
•of  the  suspension,  prolonged  beneath  the  needle.     This  mirror 

FIG.  319. 


serves  to  reflect  the  image  of  a  scale.  S,  through  a  reading  tele- 
scope, L,  by  means  of  which  deflections  of  the  needle  can  be 
observed. 

To  use  the  instrument,  the  needle  is  charged,  through  its 
suspending  wire,  to  a  constant  potential.  This  may  be  done  by 
connecting  C  with  the  knob  of  a  charged  Leyden  jar.  The 
diagonally  opposite  quadrants  are  connected  together,  and  the 
two  pairs  connected  with  the  points  whose  difference  of  potential 
is  to  be  determined.  Now  suppose  that  2  and  4  (small  diagram) 
were  of  higher  potential  than  1  and  3.  They  would  exert  a  greater 
force  upon  the  needle  than  1  and  3,  and  according  to  the  sign  of 
the  charge  on  the  needle,  would  cause  rotation  of  the  needle  in  one 
direction  or  another.  The  needle,  which  was  held  in  the  zero 
position  by  the  torsion  of  its  suspension,  would  come  to  rest  at  a 
place  where  the  force  of  torsion  was  equal  and  opposed  to  the 
electrical  force.  For  small  deflections  the  forces  are  proportional 
.to  the  tangents  of  the  angle,  i.e.,  to  the  readings  of  the  scale. 

For  very  accurate  work  many  complicated  attachments  are 
added  to  this  simple  form. 

Problems. 

y 

1.  Two  conductors,  of  capacity  10  and  15  respectively,  are  con- 
nected by  a  fine  wire  and  a  charge  of  1000  units  is  divided  between 
them :  find  the  charge  which  each  takes,  and  the  potential  to 
which  each  is  raised.  ^  lf^>  /  ^  *-<>/  V  O 


362 


ELECTRICITY     AND     MAGNETISM. 


2.  Three  spheres  of  radii  1,  2,  and  3  cm.  are  charged  to  poten- 
tials 3,  2,  and  1  respectively,  and  are  then  connected  by  a  fine 
wire  :  what  is  their  common  potential?  ^/o^/fT 

^  3.  Two  spheres,  of  capacity  2  and  3,  are  charged  respectively  to 
potentials  5  and  10  :  what  will  be  their  common  potential,  if  they 
are  placed  in  electrical  connection  ?  V 

^  4.  Two  spheres,  of  2  and  6  cm.  radius,  are  charged  respectively 
with  80  and  30  units  of  electricity  ;  compare  their  potentials.  If 
they  are  connected  by  a  fine  wire,  how  much  electricity  will  pass 
akmgit?  V"'  ^ 

*  5.  Twelve  units  of  electricity  raises  the  potential  of  a  conductor 
from  0  to  3  :  what  is  its  capacity  ?  U 


FIG.  320. 


CHAPTER   II. 

ELECTROSTATIC     INDUCTION. 

576.  Gold-leaf  Electroscope. — The   gold-leaf  electroscope 
is  a  delicate  instrument  for  detecting  the  presence  of  electric- 
ity.    It  consists  (Fig.  320)  of  a  folded  strip 
of    gold-leaf,    suspended   from   the   end   of 
a  brass  rod,  which  penetrates  the  stopper 
of  a  glass  insulating  receiver.     The  outside 
end  of   the  rod   is   provided  with  a  brass 
ball.     Whenever  a  charge  of  electricity  is 
communicated  to  the  ball   the   gold-leaves 
partake  of  it  and  diverge  from  each  other, 
because  of  the  repulsion  of  like  kinds  of 
electricity.      The  sides  of  the  receiver  are 
provided  with   strips  of  tin-foil,  which  are 
in  electrical  communication  with  the  earth 

I  through  the  base.  The  object  of  these  is  to 
prevent  the  rupturing  of  the  gold-leaves  by 
the  sudden  communication  of  too  great  a 
charge.  Upon  receiving  such  a  charge  they  diverge  and  com- 
municate it  to  the  tin-foil  and  it  escapes  thence  to  the  earth. 

577.  Phenomena  of  Induction. — Whenever  an  electrified 
body  is  approached  toward  the  brass  ball  of  an  electroscope,  it 
will  be  noticed  that,  while  it  is  even  a  great  distance  away  from 
it,  the  gold-leaves  begin  to  separate  and  show  the  presence  of 
electricity  upon  them.     This  electricity  is  the  result  of  the  presence 


EXPLANATION     OF     ATTRACTION.  363 

of  an  electrified  body  in  the  neighborhood  and  is  called  induced 
electricity.  The  process  under  which  it  was  generated  is  termed 
electrostatic  induction. 

Whenever  a  charged  conductor  is  brought  near  to  an  unchai'ged 
conductor,  and  is  separated  from  it  only  by  an  insulator,  which  in 
this  case  is  called  a  dielectric,  the  uncharged  conductor,  undergoes 
an  electrical  change.  The  side  which  is  toward  the  first  con- 
ductor is  charged  with  an  opposite  kind  of  electricity,  while  the 
remote  side  has  a  charge  of  same  kind  as  the  original  charge. 

Thus  (Fig.  320),  if  a  negatively  electrified  piece  of  hard  rubber 
be  brought  near  to  the  gold-leaf  electroscope  and  is  separated  from 
it  by  air  for  a  dielectric,  there  will  be  positive  electricity  induced 
on  the  nearer  side  of  the  electroscope,  which  is  the  ball,  and  neg- 
ative on  the  remote,  which  includes  the  gold-leaves.  The  leaves 
accordingly  diverge.  The  electricity  on  A  is  called  the  inducing 
charge,  that  in  the  electroscope  the  induced  charge. 

Whenever  an  insulated  conductor,  which  contains  the  two  kinds 
of  induced  electricity  and  is  still  under  the  influence  of  the  induc- 
ing charge,  is  connected  with  the  earth,  the  electricity  of  the  same 
kind  as  the  inducing  charge  will  escape  to  the  earth.  This  is 
because  of  the  repulsion  between  like  kinds  of  electricity.  It  is 
equally  true  whether  the  near  or  remote  side  of  the  conductor  is 
connected  to  the  earth. 

The  remaining  opposite  kind,  however,  cannot  escape  because 
of  the  attraction  exerted  by  the  original  inducing  charge.  If  now 
the  earth  connection  be  removed,  it  will  be  found  that  only  a  small 
portion  of  the  original  inducing  charge  can  escape  when  connected 
to  the  earth.  It  is  held  in  place  by  an  opposite  kind  of  electricity, 
which  it  has  itself  produced.  These  two  opposite  electricities, 
separated  by  a  dielectric,  are  said  to  be  bound,  while  electi'icity 
free  to  follow  an  earth  connection  is  called  free  electricity. 

For  illustration,  suppose  that  the  appai-atus  is  in  the  condition 
represented  in  Fig.  320.  If  the  finger  be  touched  at  C,  the  elec- 
tricity n  n  will  escape  to  the  earth  and  the  leaves  will  collapse.  The 
positive  charge  at  G  remains  bound  by  ^4's  charge.  If  now  A  be 
removed,  this  charge  will  diffuse  over  the  electroscope  and  the 
leaves  will  diverge  because  of  the  portions  which  they  receive. 

As  might  be  expected,  successive  inductions  may  be  obtained 
from  one  original  charge.  The  induced  charge  in  one  case  acts  as 
the  inducing  charge  in  a  new  induction. 

578.  Induction  Precedes  Attraction. — Whenever  a  body 
is  attracted  because  of  the  charge  of  electricity  on  another  body,  it 
is  always  subjected  to  induction  before  it  is  attracted. 


364: 


ELECTRICITY     AND     MAGNETISM. 


Thus,  if  B  (Fig.  321)  is  attracted  by  a  positive  charge  on  A, 
the  attraction  is  always  preceded  by  an  induction,  whereby  B  is 
charged  negatively  at  c  and  positively  at  d  ; 
FIG.  321.  c  is  nearer  than  d,  hence  the  attraction  be- 

tween a  and  c  is  greater  than  the  repulsion 
between  a  and  d.  Accordingly  attraction 
predominates. 


FIG. 


579.  Quantity  of  the  Induced  Elec- 
tricity.— The  total  quantity  of  electricity,  of 
the  opposite  kind  to  its  own,  which  a  charged 
body  induces  on  neighboring  bodies  is  ex- 
actly equal  to  its  own  charge.  This  was  experimentally  proved  by 
Faraday  by  means  of  an  "  ice-pail."  A  metallic  pail,  A  (Fig.  322), 
was  mounted  upon  an  insulating  support.  The  outside  of  the  pail 
was  connected  with  a  delicate  electroscope.  Into 
this  pail  was  lowered  a  positively  charged  ball, 
B,  which  was  suspended  from  an  insulating  silk 
thread.  Upon  introducing  the  ball  the  leaves  of 
the  electroscope  commenced  to  diverge,  because 
the  charge  on  B  induced  negative  electricity  on 
the  interior  of  the  pail  and  held  it  bound.  The 
positive  electricity  of  the  pail,  being  free  and  re- 
pelled, passed  partly  into  the  electroscope.  As 
the  ball  was  lowered  further  the  leaves  diverged 
more  and  more  until,  after  a  certain  depth  had 
been  reached,  a  further  descent  produced  no 
extra  divergence.  Even  when  the  ball  was 
brought  into  contact  with  the  bottom  of  the  pail 
the  leaves  remained  undisturbed  and  extended. 
Upon  removing  the  ball,  after  contact,  the  charge 
was  found  to  have  disappeared  from  it.  The 
fact  that  the  gold-leaves  were  undisturbed  by  the  contact  of  the 
ball  with  the  pail  proves  that  there  was  the  same  quantity  of 
negative  electricity  on  the  inside  of  the  pail  as  positive  electricity 
on  the  ball.  Coming  together  the  two  neutralized  each  other  and 
left  the  positive  outside  charge  undisturbed. 

580.  Condensers. — If  a  pane  of  glass  be  taken,  and  a  piece 
of  tin-foil  be  pasted  upon  the  middle  of  each  face  of  the  pane,  and 
one  piece  be  charged  positively,  the  inner  surface  of  the  other 
piece  will  receive  a  negative  charge  by  induction.  If  the  second 
piece  be  connected  with  the  earth  positive  electricity  will  escape. 
The  positive  electricity  of  the  first  tin-foil  will  attract  and  hold  the 
negative  of  the  second  bound.  If  the  connections  to  the  source 


SPECIFIC     INDUCTIVE     CAPACITY.  365 

arid  the  earth  be  now  removed,  it  will  be  found  that  hardly  any 
electricity  can  be  obtained  by  merely  touching  either  of  the  foils. 
It  may  be  said  that  each  charge  is  inducing  the  other.  It  will  be 
found  that  these  two  pieces  of  tin-foil  may  be,  when  thus  arranged, 
much  more  strongly  charged  than  either  of  them  could  possibly 
be,  if  it  were  placed  alone  upon  a  piece  of  glass  and  then  elec- 
trified. In  other  words,  the  capacity  of  a  conductor  is  greatly 
increased  when  it  is  placed  near  to  a  conductor  electrified  with 
the  opposite  kind  of  charge.  Considering  then  (Art.  570)  that  the 

potential  V  —  ^>  it  will  be  seen  that  such  a  piece  of  apparatus 
G 

can  receive  a  large  quantity  of  electricity,  Q,  without  raising  its 
potential,  V,  as  much  as  if  it  were  separated  from  all  conducting 
or  electrified  bodies.  Such  an  arrangement  is  called  a  con- 
denser. 

Condensers  are  much  used  in  practical  electricity  for  measur- 
ing quantities  of  electricity.  A  pane  of  glass,  however,  would  not 
have  a  sufficiently  large  capacity  for  technical  purposes.  Accord- 
ingly commercial  condensers  are  made  by  piling  together  alternate 
sheets  of  tin-foil  and  paraffined  paper.  The  alternate  layers  of 
tin-foil  are  connected  together.  In  this  manner  a  large  surface 
of  tin-foil  can  be  used  and  yet  not  occupy  an  inordinate  amount  of 
space.  The  capacity  of  a  condenser  varies  inversely  as  the  thick- 
ness of  the  dielectric  between  the  conducting  sheets,  and  directly  as 
the  product  of  the  area  of  .the  sheets  and  a  constant  depending 
upon  the  nature  of  the  dielectric.  This  constant  is  called  the 
specific  inductive  capacity. 

581.  Specific  Inductive  Capacity.  —  It  has  been  stated  (Art. 
563)  that  a  body  charged  with  Q  units  of  electricity  will  attract  an 
unlike  unit  of  electricity  on  a  body  which  is  r  centimetres  away 
with  a  force 


This  is  strictly  true  only  when  the  two  bodies  are  in  a  vacuum. 
It  is  very  nearly  true  when  they  are  separated  fi'om  each  other  by 
dry  air  or  any  other  gas.  That  the  expression  for  the  force  may 
be  universally  true,  whatever  be  the  dielectric  which  intervenes 
between  the  bodies,  it  must  be  modified  into  the  form 


Here  K  is  a  constant,  which  is  peculiar  to  each  substance,  and  it 
is  called  the  specific  inductive  capacity  or  the  dielectric  constant  of 
the  substance. 


366  ELECTRICITY     AND     MAGNETISM. 

The  following  is  a  table  of  specific  inductive  capacities  referred 
to  air  at  0°  C.  and  760  mm.  pressure  as  unity  : 

Air  and  most  gases 1.0 

Bisulphide  of  Carbon 2.2 

Ebonite  and  Rubber.  .• 2.3 

Paraffin 2.3 

Shellac 2.9 

Sulphur 3.7 

Glass 3.2  to  6.0 

Water 6.0 

Metals oc 

The  name  "inductive  capacity"  was  introduced  by  Faraday  in  the 
publication  of  a  series  of  experiments  upon  condensers.  He  con- 
structed a  number  of  exactly  similar  condensers,  differing  from 
each  other  only  in  the  dielectric  between  the  conducting  surfaces. 
The  dielectrics  which  he  used  were  air,  sulphur,  shellac,  and  glass. 
He  measured  the  capacities  of  these  equally  dimensioned  con- 
densers and  found  that  all  the  others  had  greater  capacities  than 
the  one  containing  air.  Remembering  that  induction  is  the  prin- 
ciple upon  which  the  condenser  works,  it  can  readily  be  seen  why 
Faraday  adopted  the  term. 

Modern  writers,  however,  in  employing  the  term  "dielectric 
constant "  indicate  their  appreciation  of  the  important  light  thrown, 
by  the  mere  existence  of  different  values  of  K,  vrpon  the  true 
nature  of  electricity.  The  fact  that  the  nature  of  the  substance 
between  two  charges  of  electricity  influences  the  magnitude  of 
their  repulsions,  disproves  the  idea  that  electrical  attractions  and 
repulsions  are  action  at  a  distance.  There  must  be  something  be- 
tween the  bodies  which  plays  a  part.  Again,  the  repulsion  of  two 
charges  of  electricity,  even  when  suspended  in  a  vacuum,  indicates 
that  this  something  must  be  the  ether.  The  ether,  then,  in  differ- 
ent dielectrics,  must  in  some  manner  be  modified  from  what  it  is 
in  a  vacuum.  This  is  known  to  be  the  case  from  the  different 
optical  properties  of  bodies. 

A  method  for  showing  directly  the  effect  of  K  in  Coulomb's  law 
has  been  constructed  by  Mayer.  Suspend  horizontally  a  silvered 
circular  disc  of  mica,  16  cms.  in  diameter,  by  a  long,  slender,  spiral 
spring.  Let  the  spring  be  in  contact  with  the  silvered  surface. 
Under  this  disc  place  another  of  metal,  which  is  movable  in  a 
vertical  direction,  and  is  connected  to  the  earth.  Charge  the 
silvered  mica  with  electricity,  using  the  spring  as  a  means  of 
connection.  If  the  distance  between  the  two  discs  is  but  two  or 
three  centimetres,  the  mica  will  be  attracted  so  as  to  extend  the 
spring.  If,  now,  a  sheet  of  paraffin  or  plate  of  glass  be  interposed 
between  the  two  discs,  the  attractive  force  will  be  observed  to 


LEYDEN    JAR.  367 

decrease.  This  is  because  K  in  the  denominator  of  the  fraction 
expressing  the  force  of  attraction  is  greater  for  glass  and  paraffin 
than  for  air. 

It  will  be  noticed,  in  the  table  given,  that  K  for  metals  and 
conductors  is  oc.  According  to  this,  4a  charged  body  cannot 
attract  a  pith-ball  through  a  metal  plate.  The  metal  is  therefore 
called  an  electric  screen.  The  screen  must  be  sufficiently  large  to 
prevent  any  attractive  force  from  working  around  its  edges,  and  it 
should  be  connected  with  the  earth  to  avoid  any  induced  electricity 
from  exerting  an  influence.  It  is  in  consideration  of  these  facts 
that  electrometers  are  surrounded  by  conducting  cages,  which  are 
connected  with  the  earth.  Any  neighboring  accidental  charges  of 
electricity  cannot  then  influence  the  electrometer  needle. 

582.  Ley  den  Jar. — A  most  convenient  form  of  condenser, 
for  demonstrative  experiments,  is  the  Leyden  jar.     It  usually  con- 
sists (Fig.  323)  of  a  glass  jar  coated  up  to  a  certain  height  on  the 
inside  and  outside  with  tin-foil.     A  brass  knob  fixed 

on  the  end  of  a  stout  brass  wire  passes  downward  FlG' 
through  an  insulating  lid  and  is  connected  by  a  chain 
with  the  interior  coating  of  tin-foil.  To  charge  the  jar, 
it  is  held,  by  the  outer  coating,  in  the  hand,  and  the 
brass  knob  is  approached  to  any  source  of  electricity. 
If  the  source  furnishes  positive  then  the  internal  coat- 
ing becomes  charged  positively,  and  this  induces  and 
binds  an  equal  amount  of  negative  electricity  on  the 
outer  coating,  while  an  equal  amount  of  positive  will  be 
rendered  free  and  will  escape  through  the  hand  to  the 
ground.  The  jar  being  now  removed,  is  said  to  be 
charged — there  exists  a  state  of  positive  potential  on  the  inside  and 
negative  potential  on  the  outside.  Botli  electricities  are  bound, 
and  neither  can  produce  effects  independently.  If,  however,  they 
be  allowed  to  come  in  connection  with  each  other  (by  joining  the 
outside  coating  and  the  ball  at  the  top  with  a  wire)  the  electricities 
rush  to  neutralize  each  other,  and  will  even  spark  across  an  air 
gap.  The  jar  is  then  said  to  be  discharged.  The  length  of  the 
-air  gap  through  which  the  spark  will  pass  depends  upon  the  differ- 
ence of  potential  between  the  inner  and  outer  coatings.  Some- 
times the  difference  of  potential  becomes  so  great,  owing  to 
carelessness  in  charging,  that  the  electricities,  in  striving  to  get 
together,  will  pierce  and  fracture  the  glass  itself. 

583.  Seat  of  the  Charge.— If  a  jar  is  made  with  a  wide,  open 
top,  and  the  coatings  movable,  then,  after  charging  the  jar,  the 
coatings  may  be  removed  and  tested  without  showing  any  trace  of 


368  ELECTRICITY     AND     MAGNETISM. 

electricity  upon  them.  If  they  be  then  replaced,  the  jar  will  be 
found  to  be  charged  as  before  removal.  Benjamin  Franklin  in- 
ferred from  this  that  the  electricity  resides  upon  the  dielectric  and 
the  coatings  serve  only  to  readily  diffuse  the  charge  over  the  surface. 

584.  Residual  Charge. — If  a  jar  be  charged,  left  for  a  time, 
discharged,  and  left  for  a  while  longer,  it  will  be  found  that,  upon 
connecting  the  two  coatings,  a  spark  may  be  obtained.     The  elec- 
tricity remaining  in  the  jar  after  the  first  discharge  is  called  the 
residual  charge.     The  amount  of  residual  charge  varies  with  the 
time  that  the  jar  has  been  left  charged.     It  also  depends  upon 
the  kind  of  dielectric  used.     No  residual  charge  has  been  found 
in  connection  with  an  air-condenser. 

585.  Modern  Theory  of  Condensers. — The  modern  ether 
theory  of  electricity  gives  a  very  satisfactory  explanation  of  con- 
densers.    According  to  this  theory  electricity  is  the  ether  itself. 
When  a  conductor  is  statically  charged,  it  is  not  the  ether  of  the 
conductor  which  constitutes  the  charge,  but  the  ether  of  the  dielec- 
tric which  surrounds  the  conductor.     More  exactly,  charging  a 
conductor  is  straining  the  ether  particles  of  the  surrounding  dielec- 
tric out  of  a  definite  position,  which  they  are  presumed  to  have  a 
tendency  to  remain  in.     Thus,  if  a  positive  charge  is  communicated 
to  the  inner  coating  of  a  Leyden  jar,  the  ether  particles  of  the  glass 
will  all  be  strained  away  from  the  inside.     All  the  particles  will  be 
strained  away  from  the  inner  coating  and  toward  the  outer  coat- 
ing.    We  may  thus  say  that  the  inner  is  positively  electrified  and 
the  outer  negatively.     We  have  but  one  ether  electrification,  but 
two  ways  of  looking  at  it — just  as  a  den^  in  a  tin  plate  may  be  con- 
vex on  one  side  and  concave  on  the  other. 

In  all  dielectrics  the  ether  particles  are  supposed  to  be  held  in 
position  by  elastic  ties  of  some  sort.     A  mechanical  analogy  is  to 
represent  the  particle  (Fig.  324)  by  a  bead  fastened  to  the  centre 
of  an  elastic  wire,  clamped  at  both  sides.     When 
FIG.  324.  subjected  to  an  electrifying  influence,  the  bead  is 

pulled  to  one  side.     Upon  releasing  the  bead,  or 
discharging  the  electricity,  two  things  are  to  be 
\  noticed.     First,  the  bead  will  not  only  go  back  to 

0  +     its  original  position,  but  will  go  beyond  it  and 
/  oscillate  a  number  of  times  before  coming  to  rest. 

According  to  this,  then,  the  spark,  at  discharging 
a  Leyden  jar,  should  be  oscillatory  and  made  up 
of  a  succession  of  small  sparks.     Such  is  the  case, 
as  has  been  shown  by  reflection   upon  a  screen  from  a  rapidly 
rotating  mirror.     Were  the  spark  a'  continuous  one,  its  reflected 


HERTZ'S     EXPERIMENTS.  369 

image  would  appear  as  a  prolonged  line.  It  appears,  however,  as 
a  dotted  line.  Secondly,  it  must  be  considered  that,  if  the  'strain 
on  the  wire  be  maintained  for  any  length  of  time,  it  will  not,  upon 
release,  immediately  return  to  its  normal  position,  but  will  assume 
a  new  one,  which  is  displaced  in  the  direction  of  the  strain  from 
the  original  position.  This  is  a  common  phenomenon  and  is 
known  as  elastic  fatigue.  The  electrical  parallel  is  the  residual 
charge.  After  the  first  discharge  the  ether  particles  have  not 
returned  to  their  normal  position.  It  requires  one  or  more  resid- 
ual discharges  before  they  return  to  that  position. 

Of  course  in  a  dielectric  we  cannot  suppose  that  each  of  the 
infinite  number  of  particles  of  ether  is  supported  upon  anything 
similar  to  an  elastic  wire,  for  the  dielectric  would  act  well  in  one 
direction  and  not  at  all  in  the  direction  of  the  length  of  the  wire. 
This  is  not  in  accordance  with  facts.  Accordingly  it  is  supposed 
that  the  dielectric  acts  as  a  mass  of  jelly,  in  which  its  ether  par- 
ticles are  suspended,  or  possibly  the  ether  is  a  jelly-like  mass  in 
itself,  lacking,  however,  the  physical  property  impenetrability. 
The  jelly  then  exerts  a  restraining  force  to  strains  exerted  in  any 
direction  upon  the  particles. 

Conductors,  on  the  other  hand,  are  supposed  to  exercise  little 
or  no  restraint  upon  the  free  movement  of  the  ether  within  them. 

Viewed  in  this  light,  when  a  positive  charge  is  communicated 
to  a  conductor,  an  attempt  is  made  to  force  more  ether  into  the 
conductor  than  it  ordinarily  has.  But  the  ether  is  absolutely 
incompressible.  Hence  room  is  made  for  the  extra  amount,  by 
pressing  the  ether  of  the  dielectric  away  from  the  conductor.  The 
extra  ether  must  have  been  taken  from  somewhere,  and  the  place 
which  it  formerly  occupied  must  be  filled.  This  is  done  by  the 
distention  of  the  remote  portions  of  the  dielectric.  It  presses  out 
of  some  conductor  an  equal  amount  of  ether.  We  can  thus  see  the 
truth  of  Faraday's  statement  that  it  is  impossible  to  charge  one 
body  alone.  Whenever  a  body  is  charged  positively,  some  other 
body  or  bodies  must  receive  an  equal  negative  charge. 

When  a  Leyden  jar  is  discharged,  by  connecting  the  two  coat- 
ings through  a  conductor,  the  ether  in  the  positive  coating  flows 
toward  the  negative  and  the  strain  on  the  dielectric  is  relieved. 
This  flow  will,  of  course,  be  vibratory,  owing  to  the  inertia  of  the 
dielectric  jelly. 

586.  Hertz's  Experiments. — Professor  Hertz  has  shown  ex- 
perimentally that  the  electrical  ether  is  wonderfully  like,  if  not  iden- 
tical with,  the  ether  presupposed  in  light.  By  rapidly  charging 
and  discharging  a  conductor  he  causes  the  ether  upon  it  to  surge 


370  ELECTRICITY     AND     MAGNETISM. 

to  and  fro.  This  agitates  the  surrounding  dielectric  ether,  and 
the  disturbances  travel  in  waves.  The  velocity  of  propagation 
he  determines  to  be  the  same  as  the  velocity  of  light.  He  has 
made  these  waves  interfere,  has  reflected  them  from  metal  mirrors, 
refracted  them  through  lenses  and  prisms  of  pitch,  and  has  pro- 
duced diffraction  effects.  He  has  shown  that  many  optical  experi- 
ments can  be  electrically  performed  by  substituting  dielectrics  for 
transparent  and  conductors  for  opaque  bodies. 

587.  Electrical  Machines. — The  method  of  rubbing  sealing- 
wax  with  fur  is  too  slow  for  the  production  of  large  quantities  of 
electricity.     Franklin   improved  upon  this  method,  employing  a 
machine,  which  rotated  a  large  glass  cylinder,  the  cylinder  being 
rubbed  by  a  silk  rubber.     But,  at  present,  machines  which  gen- 
erate electricity  by  friction  are  little  used,  recourse  being  made 
instead  to  the  principle  of  induction.     Two  machines  of  this  sort, 
most  commonly  found,  are  the  electrophorus   and  the  Holtz  ma- 
chine. 

588.  Electrophorus. — The  electrophorus  consists  of  a  cir- 
cular cake  of  resin,  sulphur,  or  vulcanized  rubber  in  a  metallic 
base,  A   (Fig.  325),  and  a  metallic  disc,  B,  having  an  insulating 

FIG  325  handle.     Stroking  with  flannel  or  fur  elec- 

trifies the  resin  negatively.  This  induces 
and  binds  positive  electricity  on  the  lower 
face  of  the  disc,  when  placed  upon  it.  Free 
negative  is  repelled  to  the  upper  face.  If 
the  finger  be  touched  to  the  disc,  while  it 
yet  remains  upon  the  resin,  the  free  neg- 
ative will  escape  to  the  earth.  Upon  then 
lifting  the  disc  it  will  be  found  charged 
with  positive  electricity.  This  operation  may  be  repeated  in- 
definitely. 

589.  Holtz  Machine. — This  machine,  illustrated  in  Fig.  326, 
consists  of  a  revolving  glass  disc,  A,  and  a  stationary  glass  disc,  B, 
both  well  coated  with  shellac  to  further  insulation.     In  front  of  A, 
and  close  to  it,  as  shown  in  the  figure,  are  two  combs  connected 
with  the  discharging  knobs  at  C.     On  the  back  of  the  disc  B, 
opposite  to  the  combs,  are  two  paper  sectors,  a  paper  tongue 
or  point  from  each  projecting,  through  an  opening  (window)  in 
the  stationary  disc,  toward  the  revolving  plate.      If  a    plate  of 
vulcanite  be  excited,  and  then  be  laid  against  one  of  the  paper 
sectors,  while  the  disc  A  is  rapidly  rotated  toward  the  point  of  ihc, 
sectors,  the  discharging  knobs  being  in  contact,  electrical  induction 


HOLTZ  MACHINE. 


371 


FIG.  326. 


•will  ensue,  and  after  a  few  moments  the  knobs  may  be  gradually 
separated  until  sparks  perhaps  12  or  20  inches  long  pass,  accord- 
ing  to  the  size  of   the  ma- 
chine.     To    produce    sparks 
of  great  density  two  Leyden 
jars,  D,  D,  with  their  inner 
coatings    connected    to    the 
discharging  knobs  and  their 
outer  coatings  connected  with 
each  other,  are  added. 

To  explain,  in  a  very  gen- 
eral way,  the  action  of  the 
machine,  let  A  (Fig.  327) 
represent  the  revolving  plate 
and  B  the  stationary  disc  be- 
hind it,  carrying  the  paper 
sectors  a  and  b.  Imagine 
the  combs  in  front  of  A,  and 
call  them  a  and  b  also,  re- 
membering that  the  sectors 
are  on  the  plate  B,  behind  A. 
If  now  a  positive  charge  .be 
communicated  to  the  sector 
a,  it  will  act  inductively  upon 
the  comb  a,  through  the  re- 
volving plate,  as  a  dielectric.  Negative  electricity  will  be  induced 
in  the  comb,  and,  if  it  were  of  proper  shape,  would  be  bound. 
But,  being  pointed,  the  negative  charge 
escapes  to  the  surface  of  A  and  is  carried 
around  with  it.  The  positive  electricity 
of  the  comb  is  repelled  by  the  process  of 
induction  to  the  discharging  knob  con- 
nected with  it.  Both  knobs  being  in 
contact  it  passes  to  the  comb  b.  From 
here  it  escapes  to  the  front  of  the  revolv- 
ing disc,  and  at  the  same  time  induces 
negative  electricity  on  the  upper  portion 
of  the  sector  b.  Because  of  the  induction 
positive  electricity  escapes  from  the  point 
of  the  sector  b  to  the  back  of  the  revolv- 
ing disc.  If  now  the  disc  be  revolved  half-way  around,  the  pos- 
itive charge  on  the  back  of  A  will  be  taken  up  by  the  point  of 
sector  a,  thus  strengthening  its  original  charge.  Sector  b,  having 
now  become  charged  negatively  increases  the  flow  of  positive  from 


372  ELECTRICITY     AND     MAGNETISM. 

comb  b,  and  sector  a,  having  its  charge  increased,  still  further 
increases  this  flow.  All  the  arrangements  conspire  to  charge  both 
the  rear  and  front  of  the  upper  half  of  A  with  positive,  and  of  the 
lower  half  with  negative,  electricity.  The  action  requires  that 
positive  electricity  shall  flow  continuously  through  the  knobs 
from  a  to  b.  When  sufficient  potential  difference  between  the 
combs  has  been  obtained,  the  discharging  knobs  may  be  separated 
and  a  spark  will  rupture  the  air. 

The  Leyden  jars  serve  to  increase  the  capacity  of  the  knobs,  and 
thus,  for  a  spark  at  a  given  potential  difference,  to  increase  the 
quantity  of  electricity  passed. 

A  modification  of  the  Holtz  machine  has  been  made  by  Wims- 
hurst.  The  two  plates,  furnished  with  numerous  sectors  of  tin- 
foil, are  rotated  in  opposite  directions.  A  full  description  is  out 
of  place  here. 

590.  Effects  of  Statical  Discharge. — Most  electrical  effects 
are  best  obtained  by  the  use  of  current  electricity.     Those  which 
require  the  high  potential  of  statical  electricity  are  the  following : 

MECHANICAL. — If  a  heavily  charged  Leyden  jar  be  made  to  dis- 
charge itself  through  a  piece  of  glass  or  card-board,  it  will,  by  the 
passage,  pierce  a  hole  through  the  piece.  In  case  card-board  is 
used,  it  will  be  found  that  there  is  a  burr  on  both  sides  of  the 
card.  This  is  because  of  the  vibratory  character  of  the  discharge. 

PHYSIOLOGICAL. — If  a  discharge  is  made  through  a  human  being, 
the  muscles  which  lie  along  its  path  will  be  strongly  contracted. 
Those  who  have  received  very  powerful  shocks  from  electrical 
discharges  say  that  the  feeling  is  as  though  all  the  muscles  had 
been  so  severely  contracted  as  to  result  in  spraining  them.  The 
action  of  the  electricity  is  through  the  medium  of  the  nerves. 

HEATING. — A  very  sudden  development  of  heat  accompanies  the 
spark  at  discharge.  This  can  be  easily  shown  by  allowing  a  spark 
to  pass  to  the  tip  of  a  gas-burner.  If  the  gas  be  turned  on,  it  will 
become  ignited.  If  gas  be  not  available,  the  spark  may  be  passed 
to  a  spoon  containing  a  few  drops  of  common  ether. 

591.  Lightning. — Water,   in   the   process  of   evaporation,   is 
supposed  to  become  electrified.     From  the  surface  of  large  bodies 
of  water  multitudes  of  small  electrified  particles  of  moisture  rise 
into  the  air,  under  the  influence  of  the  hot  sun.     These  particles 
have  a  definite  capacity  (their  radius)  and  a  definite  quantity  of 
electricity.     In   the   process   of    cloud   formation   these   particles 
come  into  drops.     Each  drop  receives  the  electricity  of  its  com- 
ponent particles  and  has  its  capacity  increased.     The  quantity  on 
the  drop  equals  the  sum  of  the  quantities  on  the  particles.     The 


MAGNETS.  373 

capacity  of  the  drop  is  less  than  the  sum  of  the  capacities  of  the 
particles.  Hence  the  potential  on  the  drop  is  greater  than  it  was 
upon  the  particles.  In  this  manner  clouds  are  formed,  having 
large  quantities  of  electricity  at  a  high  potential.  The  opposite 
kind  of  electricity  is  induced  in  the  earth,  and  the  air,  acting  as  a 
dielectric,  is  placed  under  severe  strain.  Eventually  the  strain 
becomes  too  great  and  the  air  gives  way.  The  equalization  of 
potential,  at  that  instant,  gives  what  is  termed  a  stroke  of  light- 
ning. The  intense  heat  developed  by  the  discharge  expands  the 
air,  and  the  rushing  of  cold  air  into  the  partial  vacuum,  thus 
formed,  produces  the  sound  thunder. 

Sheet  lightning,  where  a  large  surface  is  momentarily  illu- 
minated, is  but  the  reflection  from  a  cloud  of  an  invisible  true 
discharge.  / 

^ 


CH  AFTER    III. 

MAGNETISM. 

592-  Natural  Magnets.  —  The  ancients  discovered  that  a 
certain  black  stone,  abundantly  found  in  Magnesia,  had  the  prop- 
erty of  attracting  to  it  small  pieces  of  iron.  Accordingly,  from 
their  source,  they  called  these  stones  magnets.  Afterwards  they 
found  that,  when  hung  by  threads,  a  certain  part  of  each  stone 
always  pointed  north.  From  this  property  the  stone  received  the 
name  Lodestone  (leading  stone). 

593.  Artificial  Magnets. — If  a  piece  of  steel  be  rubbed  with 
a  lodestone,  it  will  be  found  to  have  acquired  the  property  of 
attraction.      Steel   artificial    magnets  are  what   are  employed   in 
experiments  in  the  laboratory. 

594.  Poles  of  a  Magnet. — If  a  steel-bar  magnet  be  rolled  in 
iron  filings  (Fig.  328), 

it  will   be   observed 

that   the  attractions 

seem    to    have    two      J|  N 

common  sources,  two 

points  near  the  ends 

of  the  bar.     These  two  points  are  called  the  poles  of  the  magnet. 

The  straight  line  connecting  the  poles  is  called  the  magnetic  axis. 

595.  Magnetic   Needle. — For  investigating  the  attractions 


374  ELECTRICITY    AND     MAGNETISM. 

of  magnets  use  is  made  of  the  magnetic  needle.     This  consists 
(Fig.  329)  of  a  light  steel  needle,  which  has  been  magnetized,  and,  by 
some  suitable  arrangement,  is  mounted 
FIG.  329.  upon  a  pivot.     It  is  capable  of  moving 

p  <fo  y      in  a  horizontal  plane  with  little  friction. 

Left  to  itself  it  will  assume  a  north  and 
south  direction.     That  end  of  it  which 
points  north  is  called  the  north  pole,  and 
the  other  end  the  south  pole.     The  com- 
passes sold  by  opticians  are  magnetic  needles,  whose  north  poles 
are  generally  more  pointed  than  their  south  poles. 

596.  Attractions   and   Repulsions. — If  a  piece  of  iron  be 
approached,   in  any  manner,   to  either  end   of   a  magnetic  nee- 
dle, the  needle  will  be  attracted  to- 
ward   the    iron.      The   same   result 

will  follow  if  either  end  of  a  magnet 
be  approached  to  an  iron  non-mag- 
netic needle.  However,  if  either  end 
of  a  magnet  be  approached  to  a  mag- 
netic needle  (Fig.  330)  attraction 
will  follow  when  the  adjacent  poles 
are  unlike,  and  repulsion  takes  place 
when  the  adjacent  poles  are  of  the 
same  kind.  Hence,  as  with  quan- 
tities of  electricity : 

Poles  of  the  same  name  repel,  and  those  of  contrary  name  attract 
each  other. 

597.  North  and  South  Poles  Inseparable.— If  any  one 
portion  of  a  piece  of  steel  be  touched  by  a  north  pole  of  a  lode- 
stone,  it  will  be  found  to  have  developed  a  south  pole.     At  the 
same  time,  however,  a  north  pole  has  been  developed  in  some 
other  part  of  the  steel.     Again,  if  a  bar  magnet  be  broken  at  a 
point  half-way  between  its  poles,  each  of  the  fragments  will  possess 
two  poles.     Successive  breaking  leaves  each  fragment  with  its  two 
different  poles.     The  end  of  a  fragment  which  had  a  pole  before 
the  rupture  retains  the  same  polarity  afterwards.     It  may  thus  be 
concluded  that  every  magnet  must  have  two  poles. 

598.  Magnetic  Induction. — When  a  bar  of  iron  is  brought 
near  to  the  pole  of  a  magnet,  though  attraction  is  the  phenomenon 
first  observed,  yet  it  is  readily  proved  that  this  attraction  results 
from  a  change,  which  is  previously  produced  in  the  iron.     Similar 
to  the  case  of  electrostatic  attraction,  the  iron  becomes  a  magnet 


COERCIVE    FORCE.  375 

by  induction  exerted  by  the  original  magnet.  By  moving  a  mag- 
netic needle  around  the  iron  it  will  be  found  that  the  end  of  the 
iron  which  is  placed  near  one  pole  of  the  magnet  becomes  a  pole 
of  the  opposite  name,  and  the  remote  end  a  pole  of  the  same  name. 
Hence  the  adjacent,  unlike  poles  of  the  iron  and  magnet,  attract 
each  other. 

The  induced  magnet  is  more  powerful  the  nearer  it  is  to  the 
inducing  magnet ;  it  is,  therefore,  greatest  when  the  two  bars  are 
in  contact. 

Soft  iron  retains  its  magnetic  properties  only  while  under  the 
influence  of  the  magnet.  Upon  removing,  it  will  be  found  to  have 
returned  to  its  neutral  state.  Had  steel,  or  impure  iron,  or  cast- 
iron  been  used,  it  would  have  been  found  to  have  retained  more  or 
less  of  the  magnetic  properties  caused  by  the  induction. 

The  inductive  action  may  be  well  seen  by  placing  iron  and 
magnet  upon  a  sheet  of  paper  and  then  sifting  iron-tilings  upon 
them.  The  filings  will  attach  themselves  to  the  iron  in  the  same 
manner  as  to  the  magnet.  If  the  magnet  be  now  withdrawn,  the 
filings  around  the  iron  will  collapse,  showing  the  loss  of  magnetic 
polarity. 

The  induced  temporary  magnet  will,  in  its  turn,  induce  tem- 
porary magnetism  in  a  second  piece  of  iron,  and  this  again  in  a 
third  piece.  The  strength,  however,  decreases  as  the  pieces  get 
farther  away  from  the  original  magnet 

If  the  north  ends  of  two  equal  magnets  be  touched  to  the 
opposite  ends  of  a  bar  of  steel,  south  poles  will  be  induced  in  both 
ends  of  the  bar.  But  we  have  seen  that  every  south  pole  must 
have  a  north  pole  with  it.  Accordingly  examination  will  reveal 
that  the  bar  has  two  coincident  north  poles  at  its  middle.  Such 
intermediate  poles  are  termed  consequent  poles. 

599.  Retentivity  or  Coercive  Force. — The  extension  of 
the  experiment  of  breaking  a  magnet  (Art.  597)  leads  to  the 
inference  that  every  particle  of  steel  is  a  magnet  in  itself.  Before 
magnetization  these  molecular  magnets  point  in  all  directions,  and 
hence  exert  no  external  magnetic  influence.  Under  the  influence 
of  induction,  however,  these  are  made  to  assume  the  same  direc- 
tion. Fig.  331  gives  an  idea  of  the  probable  arrangement  of  a 
magnetized  piece  of  steel. 

The  shaded  ends  represent  ,  Fl°-  331- 

the  south  poles  of  the  mole 
cular  magnets.  When  they 
are  all  arranged  as  in  the 

figure  the  external  effect  is  as  though  there  were  a  south  pole  at 
and  a  north  pole  at  N. 


376  ELECTRICITY     AND     MAGNETISM. 

Now  experiments  show  that  tempered  steel  is  much  more  diffi- 
cult to  magnetize  than  a  piece  of  soft  iron,  and,  after  being  once 
magnetized,  retains  its  magnetism  much  better.  It  is,  then,  rea- 
sonable to  suppose  that  the  existence  of  foreign  particles  (carbon) 
in  the  steel  hinders  and  clogs  the  turning  of  the  molecular  mag- 
nets from  their  chaotic  state  into  regular  arrangement.  Once 
arranged,  the  same  cause  prevents  a  disarrangement.  In  pure 
iron  the  hindrance  is  not  present.  This  resistance  against  a 
magnetizing  or  demagnetizing  force  is  called  retenticity  or  coercive 
force.  As  might  be  expected,  the  retentivity  is  modified  by  any- 
thing which  will  cause  the  molecules  to  vibrate,  as  hitting  sharp 
blows  with  a  hammer  or  heating  to  a  high  temperature.  A  magnet 
may  be  demagnetized  by  dropping  it  several  times  upon  the  floor. 

The  extreme  amount  of  magnetism  that  could  be  imparted  to  a 
bar  would  be  that  which  arranged  all  the  molecular  magnets  in 
the  same  direction.  The  magnet  is  then  said  to  be  saturated. 
After  removing  the  magnetizing  force,  however,  some  of  the  mole- 
cular magnets  would  of  their  own  accord  turn  from  line  and  others 
would  follow  their  example  in  time.  Hence  a  magnet  must  be 
kept  for  some  time  before  its  strength  can  be  considered  as 
constant.  Yet  a  constant  strength  may  be  obtained  by  "cooking" 
the  magnet.  It  is  saturated  and  then  placed  for  several  hours  in  a 
bath  of  steam,  removed  and  again  saturated  and  cooked.  Magnets 
treated  in  this  manner  are  said  to  remain  very  constant. 

600.  Law  of  Magnetic  Force. — Magnetic  attractions  and 
repulsions   take  place  according   to  a  law  similar  to  Coulomb's 
law  for  electrical  forces.     Two  like  isolated   magnetic   poles   of 
strengths  m  and  m ',  d  centimetres  from  each  other,  will  repel  each 
other  with  a  force  in  dynes, 

_  m  m' 
1  ~dT' 

If  the  poles  were  different,  as  m  and  —  m',  then  the  value  of  F 
would  be  negative.  A  negative  value  indicates  attraction,  and  a 
positive,  repulsion. 

The  magnetic  force  will  act  through  all  substances  except 
through  magnetic  substances,  i.e.,  those  which  are  attracted  by 
a  magnet.  No  attraction  can  take  place  through  a  large  iron 
sheet.  Such  a  piece  of  iron  is  called  a  magnetic  screen.  A  small 
magnet  suspended  in  a  hollow  iron  sphere  cannot  be  deflected  by 
an  outside  magnet. 

601.  Unit  Magnet  Pole.— If,  in  the  formula  for  the  force, 
given  in  the  previous  article,  F  and  d  be  supposed  each  equal  to 
unity,  then,  as  in  electrostatics  (Art.  563), 


LAMINATED     MAGNETS. 


377 


The  unit  magnetic  pole  is  one  that  will  repel  an  equal  like  pole, 
token  at  a  unit's  distance,  with  a  unit  force. 

Of  course  an  isolated  pole  cannot  be  obtained,  for,  in  a  magnet,  it 
is  always  accompanied  by  an  opposite  equal  pole,  and  the  algebraic 
sum  of  the  strengths  of  the  poles  of  a  magnet  always  equals  zero. 

The  poles  of  a  bar  magnet  are  the  points  from  which  all  the 
forces  may  be  considered  to  emanate.  If  the  strength  of  one  of 
these  poles  be  multiplied  by  the  distance  between  the  two  poles,  a 
quantity  results  which  is  termed  the  magnetic  moment  of  the  bar. 
In  ordinary  bar  magnets  the  pole  distance  is  about  £  of  the  total 
length  of  the  bar. 

The  magnetic  moment  of  a  bar  divided  by  its  weight  in  grams 
gives  the  specific  magnetism  of  the  substance  of  which  the  bar  is  com- 
posed. This  is  greatest  in  very  hard-tempered  steel. 
The  magnetic  moment  divided  by  the  volume  of  a 
magnet,  i.e.,  the  magnet  strength  per  unit  volume, 
is  termed  the  intensity  of  magnetization  and  is  gen- 
erally represented  by  the  letter  I. 

602.  Lifting  Power. — The  strength  of  a  mag- 
net must  not  be  confounded  with  its  lifting  power. 
The  latter  depends  upon  the  shape  of  the  magnet 
and  also  upon  the  shape  of  the  body  lifted.     A  mag- 
net bent  into  the  shape  of  a  horseshoe  (Fig.  332) 
will  lift  about  four  times  what  it  would  with  one 
end,   if   straight.     The   lifting   power   of  a   magnet 
grows  very  curiously,  if  the  load  be  gradually  in- 
creased from  time  to  time. 

603.  Laminated  Magnets. — Long,  thin  steel 
magnets  are  more  powerful  in  proportion  to  their 
weight  than  thicker  ones.     Hence  compound  mag- 
nets are  constructed,  consisting  of  thin  laminae  of 

steel  separately  magnetized  and  afterwards  bound  together  in 
bundles.  These  laminated  magnets  (Fig.  333)  are  more  powerful 

than  simple  bars  of  steel. 
The  explanation  of  this 
fact  seems  to  be  that 
ordinary  steel  magnets 
are  never  saturated,  and 

what  magnetism  they  have  results  from  molecular  arrangements 
near  the  surface.  The  compound  magnets  have  a  greater  surface 
and  are  hence  stronger.  Since  the  mutual  action  of  the  like  poles 
in  juxtaposition  tends  to  weaken  them,  the  strength  of  a  compound 
magnet  will  never  equal  the  sum  of  the  strengths  of  its  parts. 


FIG.  333. 


378 


ELECTEICITY     AND     MAGNETISM. 


That  the  magnetism  of  ordinary  bars  is  confined  to  the  surface 
has  been  shown  by  placing  the  magnet  in  acid  and  dissolving 
its  surface.  After  removal,  the  bar  showed  very  little  magnetic 
polarity. 

604.  Magnetic  Field. — Lines  of  Force. — The  space  around 
a  magnet  where  its  action  is  felt  is  termed  the  field  of  the  magnet. 
When  several  magnets  are  near  to  each  other  each  one  furnishes 
its  own  field,  and  superposed  upon  each  other  they  form  a  result- 
ant field. 

The  field  is  supposed  to  be  permeated  by  magnetic  lines  of 
force.  These  lines  represent  the  direction  along  which  the  mag- 

k    netic  attractions  and 

FIG.  334.  repulsions    act.      An 

isolated  magnetic 
pole  would  move 
along  one  of  these 
lines  under  the  at- 
traction exerted  by 
the  field  magnet.  The  general  direction  of  the  lines  of  force  of  a 
bar  magnet  are  represented  in  Fig.  334. 

The  properties  of  these  lines  can  be  best  discussed  by  con- 
sidering only  those  which  lie  in  a  given  plane  passing  through  the 
magnet.  Such  a  section  is  called  a  magnetic  spectrum.  The  spec- 
trum may  be  graphically  represented  by  placing  a  sheet  of  white 
paper  over  a  magnet  and  then  sifting  fine  iron-filings  upon  the 
paper.  A  slight  tapping  on  the  paper  will  cause  the  filings  to 
arrange  themselves  along  the  lines  of  force,  as  represented  in 

FIG.  335. 


Fig.  335.  With  a  sufficiently  large  figure  it  would  be  seen  that 
every  line  starting  from  one  pole  finds  its  way,  by  a  curved  path, 
to  the  other  pole. 


MAGNETIC    FIELD.  379 

An  isolated  north  magnetic  pole,  placed  upon  one  of  these 
lines,  would  travel  along  it  away  from  the  north  pole  of  the  field 
magnet  and  toward  the  south  pole.  An  isolated  south  pole  would 
move  in  an  opposite  direction.  For  many  reasons  it  is  desirable 
to  direct  the  lines.  Hence,  as  represented  in  Fig.  334,  the  direc- 
tion which  an  isolated  north  pole  would  move  is  taken  as  the 
positive  direction. 

Of  course  it  is  impossible  to  get  an  isolated  pole,  but  the  unde- 
sired  companion  can  be  so  far  removed  as  not  to  interfere  with 
demonstrative  experiments.  If  a  shallow  glass  dish,  containing  a 
little  water,  be  placed  over  the  magnet  and  spectrum  shown  in  Fig. 
335,  and  then  a  magnetized  sewing-needle  be  floated  in  a  vertical 
position,  by  means  of  a  small  cork,  the  lower  pole  of  the  needle 
will  be  so  much  nearer  the  magnet  than  the  upper  pole  that  it  will 
act  as  an  isolated  pole.  When  placed  over  any  line  it  will  move 
along  that  line,  however  circuitous,  until  it  reaches  the  pole  of  the 
field  magnet,  which  attracts  it.  This  experiment  is  much  more 
satisfactory  when  the  field  magnet  is  an  electro-magnet.  The 
needle  may  then  be  placed  at  any  desired  point  and  commences 
to  move  only  after  the  magnet  is  excited.  Professor  Spice  sifts  the 
filings  upon  a  glass  plate  and  projects  the  whole  experiment  from 
a  vertical  lantern. 

If  a  short  magnetic  needle  be  moved  around  a  field  whose  lines 
of  force  are  graphically  shown  by  iroix-filings,  the  needle  will  turn 
until  its  magnetic  axis  coincides  with  the  direction  of  the  lines  of 
force.  In  fact  the  filings  themselves  are  little  magnets,  made  so 
by  induction,  and  tapping  the  paper  upon  which  they  rest  serves 
the  stead  of  a  pivot.  That  the  needle  should  so  place  itself  is 
quite  natural,  for  its  north  end  tends  to  travel  in  one  direction 
and  its  south  end  in  an  opposite  direction.  The  result  is  a  couple, 
which  turns  the  needle  until  the  pulls  are  from  the  same  line  of 
force,  passing  through  the  pivot. 

l/ 

605.  Theory  of  the   Curvature  of  the  Lines. — In  any 

plane  passing  through  a  magnet,  N,  S  (Fig.  336),  let  P  be  an 
isolated  unit  north  pole.  Assume  its  distance  from  the  north 
pole  PN,  to  be  twice  as  great  as  from  the  south  pole  PS.  The 
unit  will  be  repelled  by  the  north  pole  with  a  certain  force,  which 
is  represented  in  amount  and  direction  by  the  line,  PB.  Then, 
according  to  the  law  of  magnetic  force  (Art.  600),  the  attraction 
exerted  by  the  south  pole,  which  is  only  half  as  far  away,  will  be 
four  times  as  great,  and  is  represented  in  magnitude  and  direc- 
tion by  the  line  AP.  The  resultant  of  these  two  forces  must 
be  the  diagonal,  EP,  of  the  completed  parallelogram,  and  the 


380 


ELECTRICITY     AND     MAGNETISM. 


unit  pole  would  move  along  the  line  EP.     If  elementary  paral- 
lelograms be  constructed  in  this  manner  throughout   the  field, 
FIG.  336. 


FIG.  337. 


their  diagonals,    when   connected,    will    represent    the    lines    of 
force. 

The  curvature,  then,  is  the  result  of  combined  attraction  and 
repulsion.  The  lines  of  magnetic  force  from  an  isolated  pole 
would  be  straight,  as  are  the  lines  along  which  gravitation  acts, 
and  the  law  given  in  Art.  600  is  true  for  isolated  poles  only. 

606.  Fields  from  Several  Magnets. — When  several  mag- 
nets are  in  the  same  vicinity,  the  resultant  field,  compounded 
from  the  separate  fields  of  each  magnet,  is  sometimes  curiously 

arranged.  Thus  the  field 
from  two  magnets,  whose 
north  and  south  poles  are 
opposed  to  each  other,  is 
represented  in  Fig.  337. 
A  short  magnetic  needle 
would  be  in  stable  equi- 
librium if  placed  in  any 
part  of  this  field.  Fig.  338  shows  quite  a  diffei-ent  field  where 
the  opposed  poles  are  like  named.  A  needle  moved  about  this 
field  would  suddenly  turn 
half  round  on  its  axis  at 
the  moment  of  ci-ossing  the 
line  cd.  When  the  pivot 
is  exactly  upon  cd,  the 
needle's  south  pole  is  at- 
tracted equally  in  opposite 
directions  by  the  two  ex- 
posed north  poles.  In  the  same  manner  its  north  pole  is  repelled. 


FIG.  338. 


STRENGTH     OF     THE     FIELD.  381 

The  result  is  that  the  needle  places"  itself  so  that  its  axis  is  per- 
pendicular to  the  axis  of  the  field  magnets. 

Much  may  be  learned  by  experimenting  with  iron  filings  on 
vaiiously  compounded  fields. 

607.  Strength  or  Intensity  of  Magnetic  Field. — It  may 

be  reasonably  supposed  that  each  line  of  force  exerts,  along  its 
length,  a  given  amount  of  force.  Hence  a  piece  of  iron,  which 
was  traversed  by  several  lines,  would  be  more  powerfully  attracted 
than  if  traversed  by  a  fewer  number.  Thus  a  magnet  pole  of 
definite  size,  placed  near  to  the  pole  of  the  field  magnet,  would  be 
attracted  with  more  force  than  at  a  distance,  for  the  lines  are 
closer  together  near  the  poles  of  the  field  magnet.  The  number 
of  lines  of  force,  then,  which  penetrate  a  given  area,  determines 
the  relative  force  exerted  by  the  field  at  that  place.  This  is  termed 
the  strength  or  intensity  of  the  field. 

In  order  to  compare  the  strengths  of  different  fields,  it  is  neces- 
sary to  have  a  unit  of  strength.  Hence 

A  magnetic  field  of  unit  strength  is  one  which  exerts  a  unit  force 
(dyne)  upon  a  free  unit  magnet  pole. 

608.  Determination  of  the   Strength  of  a  Field.— If  a 

magnet,  suspended  by  a  fibre,  be  placed  in  magnetic  fields  of 
different  strengths,  it  will  oscillate  for  a  long  time,  and  the  times 
-of  oscillation  will  be  shorter  the  stronger  the  field.  This  is  paral- 
lel to  the  case  of  pendulum  vibrations.  The  pendulum  vibi'ates 
because  of  the  force  exerted  by  gravity  and  because  of  its  inertia. 
Gravity  pulls  its  centre  of  gravity  as  near  as  possible  to  the  earth, 
and  inertia  carries  it  beyond  this  position.  If  the  force  of  gravity 
were  increased,  the  pendulum  would  vibrate  quicker.  In  the  case 
of  a  magnet,  the  force  of  the  field  takes  the  place  of  gravity. 
Now,  just  as  the  force  of  gravity  can  be  measured  by  the  time  of 
oscillation  of  a  given  pendulum  (Art.  163),  so  the  strength  of  a 
magnetic  field  can  be  measured  by  the  time  of  oscillation  of 
a  given  magnet. 

If  t  =  the  time  taken  by  the  magnet  in  passing  from  one 
turning-point  to  the  other  in  an  oscillation,  K  =  the  moment  of 
inertia  of  the  magnet  (Art.  160),  M  =  the  magnetic  moment  of  the 
magnet,  then  the  strength  of  the  field 

JK 

H  ~  FJi 

When  the  same  magnet  is  used  K  and  M  are  constant,  hence 

H^~  oc  n\ 
where  n  =  the  number  of  single  vibrations  in  a  second.     Then  if 


382  ELECTRICITY     AND     MAGNETISM. 

a  given  magnet  vibrates  n  and  n'  times  per  second  in  two  fields  of 
strengths  H  and  H', 

H^  _  tf_ 

H'  ~  n'* 

If  the  values  of  M  and  K  are  known  or  determined,  then  the  first 
equation  gives  the  absolute  strength  of  the  field,  provided  all  the 
quantities  are  expressed  in  proper  units. 

609.  Hysteresis. — If  a  piece  of  iron  be  placed  in  a  magnetic 
field,  it  will  have  two  opposite  poles  induced  in  it  whose  strengths 
depend  upon   the  strength  of  the  field.     If  the  strength  of  the 
field  be  varied  from  zero  to  a  maximum,  and  then  to  zero  again, 
there  will  be  two  times  when  the  field  will  have  a  definite  strength 
— once  when  the  field  is  growing  stronger  and  again  when  it  is 
decreasing  in  strength.     The  strengths  of  the  induced  poles  in  the 
iron  are  different  in  these  two  equal  fields.     They  will  be  less  ill 
the  increasing  field  than  in  the  decreasing.     The  strength  of  the 
iron's  poles  depends  upon  the  iron's  previous  history.     The  iron 
has  a  tendency  to  remain  in  its  previous  condition  and  behind  the 
field's  requirements.     This  peculiarity  of  the  iron  is  termed  by 
Ewing  static  hysteresis. 

If  iron  be  placed  in  a  magnetic  field  of  constant  strength,  it 
will  require  a  certain  time  before  its  induced  poles  assume  con- 
stant strengths.  To  this  property  of  the  iron  Ewing  gives  the 
name  viscous  hysteresis. 

610.  Number  of  Lines  of  Force  from  a  Given  Pole. — It 
is  convenient  to  consider  the  number  of  lines  of  force  passing 
through  a  given  area  in  a  field  as  the  measure  of  the  strength  of 
the  field.     Each  line  may  be  supposed  to  exert  a  dyne  of  force  on 
a  unit  pole  pierced  by  it.     The  given  area  is  a  square  centimetre 
and  is  placed  so  as  to  be  perpendicular  to  the  lines  of-  force.     A 
unit  field  would  then  have  one  line  passing  through  a  square  centi- 
metre. 

Suppose  now  that,  around  a  unit  magnet  pole,  we  conceive  a 
spherical  shell  of  one  centimetre  radius.  From  the  definition  of  a 
unit  pole  (Art.  601)  we  know  that  the  enclosed  pole  exerts,  on 
another  unit  pole,  a  dyne  of  force  at  every  point  on  this  shell. 
The  strength  of  the  field,  then,  at  all  these  points,  is  unity.  Hence 
every  centimetre  of  it  is  pierced  by  one  line  of  force.  But  the 
whole  surface  of  the  sphere  of  unit  radius  contains  4  TT  centi- 
metres. The  unit  pole,  therefore,  sends  off'  4  TT  lines  of  force.  An 
enclosing  surface  of  any  size  would  be  pierced  by  the  same  number 
of  lines. 


MAGNETIC     SUSCEPTIBILITY. 


383 


If  the  strength  of  the  pole  were  2  units,  it  would  send  off  8  TT 
lines  ;  or,  in  general, 

A  magnet  pole,  of  strength  m,  sends  out  4  TT  m  lines  of  force. 
This  conception  of  the  magnetic  lines  has  recently  developed  into 
many  important  theoretical  conclusions,  which  have  equally  im- 
portant practical  applications. 

With  a  real  magnet,  having  two  poles,  it  is  important  to  re- 
member that  the  lines  of  induction,*  starting  out  from  one  pole, 
finally  arrive  at  the  other  pole  and  thence  pass  through  the  magnet 
itself.  Hence  the  number  passing  through  a  section  of  the  mag- 
net lying  midway  between  the  poles  is  4  TT  m. 


611.  Magnetic  Susceptibility. — If  two  magnets,  with  their 
opposite  poles  opposed  to  each  other,  be  arranged  along  a  common 
axis,  and  if  the  lines  of  induction  be  made  visible  by  iron-filings,  the 
resulting  spectrum  will  be  as  in  Fig.  339.  If,  now,  a  piece  of  iron 


FIG.  339. 


FIG.  340. 


N  feV 


is  placed  between  the  poles,  the  field  alters  and  will  give,  e.g.,  the 
spectrum  shown  in  Fig.  340.  The  lines  are  much  more  numerous 
where  the  iron  is  than  they  were  before  it  was  placed  there.  Had 
a  piece  of  brass  been  used  instead  of  iron,  the  field  would  have 
remained  undistorted.  A  piece  of  cobalt  would  have  produced 
some  distortion,  but  not  as  much  as  the  iron. 

The  cause  of  the  additional  lines  is  that  the  iron  has,  under  the 
influence  of  the  field,  become  an  induced  magnet  and  has  added 
its  lines  to  those  already  in  the  field. 

If  the  field  magnets  were  made  stronger,  they  would  send  out 
more  lines,  and  the  iron  would  become  more  strongly  magnetized 
and  would  also  send  out  more  lines.  Thus,  supposing  that  the 
iron  does  not  become  saturated,  the  strength  of  its  pole  depends 
upon  the  strength  of  the  field  and  upon  the  volume  of  the  iron. 
Suppose  that  the  strength  of  the  induced  pole  of  a  rectangular 
prism  of  iron  is  m  ;  that  the  pole  length  equals  the  physical 


*  A  line  of  induction  differs  from  a  line  of  force  in  that  it  does  not  change 
its  direction  on  its  return  through  the  body  of  the  magnet. 


384r  ELECTRICITY     AND     MAGNETISM. 

length  of  the  iron,  I ;  that  the  cross-section  of  the  iron  is  s.     Then 
the  intensity  of  magnetization  (Art.  601) 
.  _  m  I  _  m 
si          s 

If  the  cross-section  s  be  one  centimetre,  then  I  =  m,  or  the  inten- 
sity of  magnetization  is  equal  to  the  strength  of  the  induced  pole. 
It  has  been  found  that  if  the  strength  of  the  field  equals  H, 

I  =  &H, 

where  k  is  a  constant  called  the  magnetic  susceptibility.  It  depends 
upon  the  kind  of  iron  or  other  substance  placed  in  the  field.  For 
iron,  nickel,  and  cobalt  the  value  is  positive ;  for  vacuum,  air,  and 
most  gases  is  practically  zero,  and  for  bismuth,  antimony,  and 
phosphorus  it  is  negative,  though  extremely  small. 

612.  Magnetic  Permeability. — In  most  all  of  the  practical 
problems  on  magnetic  induction  it  is  desirous  to  know  the  total 
number  of  lines  which  pass  through  the  iron  suffering  induction. 
In  the  iron  prism  of  the  preceding  article  the  total  number 
traversing  it  is  made  up  of  two  parts :  4  TT  \  —  4  -JT  k  H  lines  from 
the  induced  pole  and  H  lines  from  the  original  field.  Represent- 
ing this  sum  by  B,  we  have 

B  =  H  +  47T&H 
=  (1  +  4  IT  k)  H. 
It  is  customary  to  place  1  -+-  4  TT  k  =  p.,  whence 

B  =/*H. 

Since  /*.  involves  k,  it  depends  upon  the  character  of  the  substance 
under  induction.  For  air  and  gases  it  is  unity;  for  iron,  etc., 
greater  than  unity  (sometimes  reaching  16,000),  and  for  bismuth, 
etc.,  less  than  unity. 

As  B  represents  the  number  of  lines  that  pass  through  a  square 
centimetre  of  iron,  and  H  the  number  through  air,  then  the  iron 
may  be  said  to  conduct  magnetic  lines  /A  times  better  than  air. 
From  consideration  in  this  light  /*  has  received  the  name  magnetic 
permeability. 

The  magnetic  permeability  of  a  substance  is  its  relative  con- 
ductivity for  magnetic  lines  of  force  as  compared  ivith  vacuum  (or 
air)  as  a  standard. 

The  equations  connecting  B,  H,  I,  /A,  and  k,  which  have  been 
given  are  true  whatever  be  the  cross-section  of  the  iron  under 
induction.  The  assumption  of  a  square  centimetre  cross-section 
is  for  simplification  only. 

613.  The  Magnetic  Circuit. — In  the  construction  of  most 
electro-magnetic  apparatus  it  is  of  utmost  importance  that  as  much 
as  possible  of  the  field  of  the  magnetizing  agent  shall  be  occupied  by  a 


DIAMAGXETISM.  385 

anbstance  of  great  permeability  such  as  iron.  For  instead  of  having 
merely  the  lines  which  can  be  sent  through  air  by  the  agent  \\e 
can  just  as  well  have  the  additional  ones  from  the  iron.  Of  course 
an  air  gap  must  be  left  somewhere  in  the  circuit  of  the  lines  in 
order  to  introduce  the  body  to  be  acted  upon.  But  this  gap 
should  be  as  small  as  possible  if  a  maximum  effect  be  desired. 

If  the  opposite  poles  of  two  straight  electro-magnets  be  caused 
to  attract  a  piece  of  iron,  the  iron  fills  in  one  gap,  but  the  lines 
from  the  other  ends  pass  through  the  air.  The  force  of  the 
original  attraction  would  be  much  increased  if  the  extreme  ends 
were  connected  by  an  iron  bar.  This  last  bar  sends  its  additional 
lines  through  the  magnets  and  increases  the  force. 


614.  Paramagnetism  and  Diamagnetism. —  Substances 
which  have  a  permeability  greater  than  1  (that  of  vacuum)  as  iron, 
steel,  nickel,  cobalt,  etc.,  are  attracted  by  a  magnet  and  tend  to 
move  toward  it.  If  not  allowed  to  move  toward,  but  allowed  to 
rotate,  they  will  tend  to  set  themselves  axially  with  the  lines  of 
induction.  These  are  called  paramagnetic  substances. 

Substances  of  permeability  less  than  unity  show  the  opposite 
tendencies.  They  are  repelled  by  magnets  and  set  themselves 
perpendicular  to  the  lines  of  force.  They  are  bismuth,  antimony, 
phosphorus,  etc.  Without  making  use  of  the  term  permeability 
we  may  say : 

Those  substances  which  are  attracted  by  a  magnetic  pole,  or  which 
in  a  magnetic  field  tend  to  move  from  places  of  less  to  places  of 
greater  intensity,  are  called  Paramagnetic. 

Those  substances  which  are  repelled  by  either  pole  indifferently, 
or  which  move  from  places  of  greater  intensity  to  places  of  less    • 
intensity  in  the  field,  are  called  Diamagnetic. 

In  order  to  explain  the  phenomena  of  paramagnetism  and  dia- 
magnetism  we  have  to  consider  that  the  movable  parts  of  a  mag- 
netic circuit  strive  to  adjust  themselves  so  that  the  maximum  lines 
of  induction  shall  pass  through  the  circuit.  Paramagnetic  sub- 
stances are  thus  drawn  into  the  circuit  and  place  themselves 
longitudinally  with  the  lines,  while  diamagnetic  substances  act 
in  an  opposite  manner,  the  air  furnishing  more  lines  than  if  they 
should  displace  it. 

The  repulsion  of  diamagnetic  substances  is  hard  to  illustrate 
before  a  large  audience.  A  huge  electro-magnet  may  be  made  to 
slightly  repel  a  piece  of  bismuth  suspended  on  a  long,  delicate 
fibre.  Better  results  can  be  obtained  by  approaching  a  large 
piece  of  bismuth  to  one  of  the  needles  in  an  astatic  magnetom- 
eter (Art.  625). 


386 


CHAPTER    IV. 

TERRESTRIAL      MAGNETISM. 

615.  The   Earth  a  Magnet. — If  a  needle  is  carried  round 
the   earth   from   north    to  south,   it  takes  approximately  all   the 
positions  in  relation  to  the  earth's  axis  which  it  assumes  in  rela- 
tion to  a  magnetic  bar,  when  carried  round  it  from  end  to  end. 
At  the  equator  it  is  nearly  parallel  to  the  axis,  and  it  inclines 
at  larger  and  larger  angles   as   the   distance   from   the   equator 
increases ;  and  in  the  region  of  the  poles  it  is  nearly  in  the  direc- 
tion of  the  axis.     The  earth  itself,  therefore,  may  be  considered  a 
magnet,  since  it  affects  a  needle  as  a  magnet  does,  and  also  induces 
the  magnetic  state  on  iron.     But  it  is  necessary,  on  account  of  the 
attraction  of  opposite   poles,  to  consider   the   northern   part  of 
the  earth  as  being  like  the  south  pole  of  a  needle,  and  the  south- 
ern part  like  the  north  pole. 

616.  Declination   of   the    Needle.  — When  the  needle  is 
balanced  horizontally,  and  free  to  revolve,  it  does  not  generally 
point  exactly  north  and  south ;  and  the  angle  by  which  it  deviates 
from  the  meridian  is  called  the  declination.     A  vertical  circle  coin- 
cident with  the  direction  of  the1  needle  at  any  place  is  called  the 
magnetic   meridian.      As   the   angle    between    the   magnetic   and 
the    geographical   meridians   is   generally   different   for   different 
places,  and  also  varies  at  different  times  in  the  same  place,  the 
word  variation  expresses  these  changes  in  declination,  though  it 
is  much  used  as  synonymous  with  declination  itself. 

The  force  which  causes  the  needle  to  set  in  the  magnetic 
meridian  is  merely  directive. 

If  the  needle  be  weighed  before  it  is  magnetized  and  again 
after  it  has  been  made  a  magnet,  no  change  of  weight  can  be 
detected,  proving  that  the  earth's  attraction  for  one  pole  is  exactly 
equal  to  its  repulsion  of  the  other.  This  may  also  be  shown  by 
attaching  a  magnet  to  a  cork  and  thus  floating  it  upon  water.  It 
will  set  in  the  magnetic  meridian  but  will  show  no  tendency  to 
move  across  the  water  toward  the  north,  nor  in  any  other  direc- 
tion. This  effect  is  due  to  the  earth's  uniform  magnetic  field. 
The  magnetic  pole  of  the  earth  being  practically  at  an  infinite 
distance,  the  forces  of  attraction  and  repulsion,  being  equal,  con- 
stitute a  couple. 

617.  Isogonic  Curves. — This  name  is  given  to  a  system  of 
lines  imagined  to  be  drawn  through  all  the  points  of  equal  decli- 


ISOGONIC    CURVES.  387 

nation  on  the  earth's  surface.  We  naturally  take  as  the  standard 
line  of  the  system  that  which  connects  the  points  of  no  declina- 
tion, or  the  isogonic  of  0°  (Fig.  341).  Commencing  at  the  north 

FIG.  341. 


pole  of  dip,  about  Lat.  70°,  Lon.  96°,  it  runs  in  a  general  direc- 
tion E.  of  S.,  through  Hudson's  Bay,  across  Lake  Erie,  and  the 
State  of  Pennsylvania,  and  enters  the  Atlantic  Ocean  on  the  coast 
of  North  Carolina.  Thence  it  passes  east  of  the  West  India 
Islands,  and  across  the  N.  E.  part  of  South  America,  pursuing  its 
course  to  the  south  polar  regions.  It  reappears  in  the  eastern 
hemisphere,  crosses  Western  Australia,  and  bears  rapidly  westward 
across  the  Indian  Ocean,  and  then  pursues  a  northerly  course 
across  the  Caspian  Sea  to  the  Arctic  Ocean.  There  is  also  a 
detached  line  of  no  declination,  lying  in  eastern  Asia  and  the 
Pacific  Ocean,  returning  into  itself,  and  inclosing  an  oval  area  of 
40°  X.  and  S.  by  30°  E.  and  W.  Between  the  two  main  lines  of 
no  declination  in  the  Atlantic  hemisphere,  the  declination  is  west- 
ward, marked  by  continued  lines  in  Fig.  341 ;  in  the  Pacific 
hemisphere,  outside  of  the  oval  line  just  described,  it  is  eastward, 
marked  by  dotted  lines.  Hence,  on  the  American  continent,  in 
all  places  east  of  the  isogonic  of  0°,  the  north  pole  of  the  needle 
declines  westward,  and  in  all  places  west  of  it,  the  north  pole 
declines  eastward ;  on  the  other  continent  this  is  reversed,  as 
shown  by  the  figure. 

Among  other  irregularities  in  the  isogonic  system,  there  are 
two  instances  in  which  a  curve  makes  a  wide  sweep,  and  then 
intersects  its  own  path,  while  those  within  the  loop  thus  formed 
return  into  themselves.  One  of  these  is  the  isogonic  of  8°  40'  E., 
which  intersects  in  the  Pacific  Ocean  west  of  Central  America ; 
the  other  is  that  of  22  13'  W.,  intersecting  in  Africa. 


388  ELECTRICITY     AND     MAGNETISM. 

In  the  northeastern  part  of  the  United  States  the  declination 
has  long  been  a  few  degrees  to  the  west,  with  very  slow  and 
somewhat  irregular  variations. 

618.  Secular   and    Annual   Variation. — The    declination 
of  the  needle  at  a  given  place  is  not  constant,  but  is  subject  to  a 
slow  change,  which  carries  it  to  a  certain  limit  on  one  side  of  the 
meridian,  when  it  becomes  stationary  for  a  time,  and  then  returns, 
and  proceeds  to  a  certain  limit  on  the  other  side  of  it,  occupying 
two  or  three  centuries  in  each  vibration.     At  London,  in  1580,  the 
declination  was  11£°  E. ;  in  1657,  it  was  0° ;  after  which  time  the 
needle  continued  its  western  movement  till  1818,  when  the  decli- 
nation was  24£°   W. ;    since   then   the  needle   has  been  moving 
slowly  eastward,  and  in  1879,  at  Kew,  the  declination  was  19° 
7'  west. 

The  entire  secular  vibration  will  probably  last  more  than  three 
centuries.  The  average  variation  from  1580  to  1818  was  9'  10" 
annually.  But,  like  other  vibrations,  the  motion  is  slowest  to- 
ward the  extremes. 

There  has  also  been  detected  a  small  annual  variation,  in  which 
the  needle  turns  its  north  pole  a  few  minutes  to  the  east  of  its 
mean  position  between  April  and  July,  and  to  the  west  the  rest 
of  the  year.  This  annual  oscillation  does  not  exceed  15  or  18 
minutes.  •  -- 

619.  Diurnal  Variation. — The  needle  is  also  subject  to  a 
small  daily  oscillation.      In  the  morning   the  north  end  of   the 
needle  has  a  variation  to  the  east  of  its  mean  position  greater  than 
at  any  other  part  of  the  day.     During  winter  this  extreme  point 
is  attained  at  about  8  o'clock,  but  as  early  as  7  o'clock  in  the  sum- 
mer.    After  reaching  this  limit  it  gradually  moves  to  the  west, 
and  attains  its  extreme  position  about  3  o'clock  in  winter,  and 
1  o'clock  in  summer.     From  this  time  the  needle  again  returns 
eastward,  reaching  its  first  position  about  10  P.M.,  and  is  almost 
stationary  during  the  night.     The  whole  amount  of  the  diurnal 
variation  rarely  exceeds  12  minutes,  and  is  commonly  much  less 
than  that.     These  diurnal  changes  of  declination  are  connected 
with  changes  of  temperature,  being  much  greater  in  summer  than 
in  winter.     Thus,  in  England  the  mean  diurnal  variation  from 
May  to  October  is   10  or  12  minutes,   and   from  November  to 
April  only  5  or  6  minutes. 

620.  Magnetometer.  —  In   determining   and  observing   the 
variation   of    the   declination   use   is   made   of  a   magnetometer. 


MAGNETOMETERS. 


389 


FIG.  342. 


Fig.  342  represents  such  an  instrument.  It  consists  of  a  mag- 
netized ring  surmounted  by  a  circular  mirror,  both  being  sus- 
pended by  a  silk  fibre.  The  poles  of  the 
ring  are  at  the  sides  and  the  plane  of 
the  ring,  when  at  rest,  coincides  with  the 
plane  of  the  magnetic  meridian.  Any  vari- 
ation of  the  meridian  is  followed  by  a  move- 
ment of  the  ring.  The  mirror,  being  con- 
nected with  the  ring,  moves  also.  This 
small  movement  may  be  magnified  and 
observed  by  means  of  a  telescope  and  scale. 
The  image  of  the  scale  is  reflected  from 
the  mirror  into  the  telescope. 

Surrounding  the  ring  magnet  is  a 
hollo  wed-out  piece  of  pure  copper.  This 
brings  the  magnet  quickly  to  rest  by  means 
of  the  electrical  currents  induced  in  it 
(Art.  671)  by  the  moving  magnet. 

Magnetometers  are  also  used  in  deter- 
mining the  magnetic  moment  of  bar  magnets. 


FIG.  343. 


621.  Dip  of  the   Needle.— A  needle 

first   balanced   on   a  horizontal   axis,   and   then   magnetized   and 
placed  in  the  magnetic  meridian,  assumes  a  fixed  relation  to  the 

horizon,  one  pole  or  the 
other  being  usually  de- 
pressed below  it. 

The  axis  of  the  needle 
must  be  placed  very  ac- 
curately at  right  angles  to 
the  plane  of  the  magnetic 
meridian,  or  false  indications 
will  be  given ;  if  the  axis  of 
suspension  were  placed  in 
the  plane  of  the  meridian 
the  angle  of  depression 
would  be  90°  at  all  places 
on  the  earth's  surface. 

The  angle  of  depression  is 
called  the  dip  of  the  needle. 
Fig.  343  represents  the  dip- 
ping-needle, with  its  adjust- 
ing screws  and  spirit-level ; 
and  the  depression  may  be 
the  graduated  scale.  After  the  horizontal  circle  ?n  is 


390  ELECTRICITY     AND     MAGNETISM. 

levelled  by  the  foot-screws,  the  frame  A  is  turned  horizontally  till 
the  vertical  circle  M  is  in  the  magnetic  meridian.  For  north 
latitudes,  the  north  end  of  the  needle  is  depressed,  as  a  in  the 
figure. 

622.  Isoclinic  Curves. — A  line  passing  through  all  points 
where  the  dip  of  the  needle  is  nothing,  i.e.,  where  the  dipping 
needle  is  horizontal,  is  called  the  magnetic  equator  of  the' earth. 
It  can  be  traced  in  Fig.  344  as  an  irregular  curve  around  the 

FIG.  344. 


earth  in  the  region  of  the  equator,  nowhere  departing  from  it 
more  than  about  15°.  At  every  place  north  of  the  magnetic 
equator  the  north-seeking  pole  of  the  needle  descends,  and  south 
of  it  the  south-seeking  pole  descends  ;  and,  in  general,  the  greater 
the  distance,  the  greater  is  the  dip.  Imagine  now  a  system  of 
lines,  each  passing  through  all  the  points  of  equal  dip ;  these  will 
be  nearly  parallel  to  the  magnetic  equator,  which  may  be  regarded 
as  the  standard  among  them.  These  magnetic  parallels  are  called 
the  isodinic  curves  ;  they  somewhat  resemble  parallels  of  latitude, 
but  are  inclined  to  them,  conforming  to  the  oblique  position  of 
the  magnetic  equator.  In  the  figure,  the  broken  lines  show  the 
dip  of  the  south  pole  of  the  needle  ;  the  others,  that  of  the  north 
pole.  The  points  of  greatest  dip,  or  dip  of  90°,  are  culled  the 
poles  of  dip.  There  is  one  in  the  northern  hemisphere,  and  one  in 
the  southern.  The  north  pole  of  dip  was  found,  by  Captain  James 
C  Ross,  in  1831,  to  be  at  or  very  near  the  point,  70°  14'  N. ;  96° 
40'  \V.,  marked  x  in  the  figure.  The  south  pole  is  not  yet  so  well 
determined. 

At  the  poles  of  dip  the  horizontal  needle  loses  all  its  directive 
power,  because  the  earth's  magnetism  tends  to  place  it  in  a  verti- 
cal line,  and,  therefore,  no  component  of  the  force  can  operate  in 


HORIZONTAL     INTENSITY.  391 

a  horizontal  plane.  The  isogonic  lines  in  general  converge  to  the 
two  dip-poles ;  bat,  for  the  reason  just  given,  they  cannot  be 
traced  quite  to  them. 

The  dip  of  the  needle,  like  the  declination,  undergoes  a  varia- 
tion, though  by  no  means  to  so  great  an  extent. 

In  1576,  the  date  of  its  discovery,  the  dip  at  London  was 
71°  50';  it  increased  to  a  maximum  of  74°  42'  in  1723,  since 
which  time  it  has  gradually  decreased.  In  1879  the  dip  at  Kew 
was  67°  42'. 

lu  the  course  of  250  years,  it  has  diminished  about  five  degrees 
in  London.  In  1820  it  was  about  70°,  and  diminishes  from  two 
to  three  minutes  annually. 

Since  the  dip  at  a  given  place  is  changing,  it  cannot  be  sup- 
posed that  the  poles  are  fixed  points  ;  they,  and  with  them  the  entire 
system  of  isoclinic  curves,  must  be  slowly  shifting  their  locality. 

623.  Intensity  of  the  Earth's  Magnetism. — The  axis  of 
the  dip-needle,  when  placed  in  the  magnetic  meridian,  coincides 
in  direction  with  the  lines  of  force  of  the  earth's  magnetic  field. 
The  magnetic  force,  then,  acts  in  this  inclined  direction.  In  most 
magnetic  determinations,  however,  the  needle  employed  swings  in 
a  horizontal  plane,  and  the  force  exerted  upon  it  by  the  earth  is 
only  that  portion  of  its  total  force  which  acts 
in  a  horizontal  direction.  This  horizontal  com- 
ponent of  the  strength  of  the  field  is  called  the 
horizontal  intensity  of  the  earth's  magnetism.  Let 
/  (Fig.  345)  represent  the  strength  of  the  earth's 
field  along  the  lines  of  force,  i.e.,  along  the 
axis  of  the  dip-needle,  d  =  angle  of  dip,  then 
h  =  the  horizontal  intensity.  From  the  diagram  it  is  seen  that 

h  —  I  cos  d. 

The  determination  of  the  horizontal  intensity  is  effected  after 
the  manner  described  in  Art.  608.  Its  values  for  places  in  North 
America  are  given  in  the  following  table  : 

HORIZONTAL  INTENSITY.     (C.  G.  S.  UNITS.) 

Boston 0.172 

Cleveland   0.184 

Chicago  0.184 

Halifax 0.159 

Montreal   0.147 

New  York 0.184 

New  Orleans 0.281 

Niagara 0. 167 

Philadelphia 0. 194 

San  Francisco 0.255 

Washington 0.200 


392  ELECTRICITY     AND     MAGNETISM. 

624.  Isodynamic  Curves. — An  inspection  of  the  table  just 
given  shows  that  the  horizontal  intensity  increases  as  we  near  the 
equator.  The  strength  of  the  earth's  field  in  the  direction  of  its 
lines  of  force,  however,  decreases  on  nearing  the  equator,  as  might 
be  expected,  the  equator  being  farthest  from  the  poles.  After 
ascertaining,  by  actual  observation,  the  intensity  of  the  magnetic 
force  in  different  parts  of  the  earth,  lines  are  supposed  to  be  drawn 
through  all  those  points  in  which  the  force  is  the  same  ;  these 
lines  are  called  isodynamic  curves,  represented  in  Fig.  346.  These 

Fro.  346. 


also  slightly  resemble  parallels  of  latitude,  but  are  more  irregular 
than  the  isoclinic  lines.  There  is  no  one  standard  equator  of 
minimum  intensity,  but  there  are  two  very  irregular  lines  sur- 
rovmdiiig  the  earth  in  the  equatorial  region,  in  some  places  almost 
meeting  each  other,  and  in  others  spreading  apart  more  than  two 
thousand  miles,  on  which  the  magnetic  intensity  is  the  same. 
These  two  are  taken  as  the  standard  of  comparison,  because  they 
are  the  lowest  which  extend  entirely  round  the  globe.  The  inten- 
sity on  them  is  therefore  called  unity,  marked  1  in  the  figure.  In 
the  wide  parts  of  the  belt  which  they  include — lying  one  in  the 
southern  Atlantic,  and  the  other  in  the  northern  Pacific  oceans — 
there  are  lines  of  lower  intensity  which  return  into  themselves, 
without  encompassing  the  earth.  In  approaching  the  polar  regions, 
both  north  and  south,  the  curves,  retaining  somewhat  the  form  of 
the  unit  lines,  are  indented  like  an  hour-glass,  as  those  marked  1.7 
in  the  figure,  and  at  length  the  indentations  meet,  forming  an 
irregular  figure  8;  and  at  still  higher  latitudes,  are  separated  into 
two  systems,  closing  up  around  two  poles  of  maximum  intensity. 
Thus  there  are  on  the  earth  four  poles  of  maximum  intensity,  two 
in  the  northern  hemisphere  and  two  in  the  southern.  The  Ameri- 


ASTATIC    NEEDLES.  393 

can  north  pole  of  intensity  is  situated  on  the  north  shore  of  Lake 
Superior.  The  one  on  the  eastern  continent  is  in  northern  Siberia. 
The  ratio  of  the  least  to  the  greatest  intensity  on  the  earth  is 
about  as  0.7  to  1.9;  that  is,  as  1  to  2-f.  In  the  figure,  intensities 
less  than  1  are  marked  by  dotted  lines. 

625.  Variation  in  the  Strength  of  the  Earth's  Field. — 
Astatic  Needles. — The  intensity  of  the  earth's  magnetism  is 
constantly  changing.  These  changes  consist  in  small  fluctuations 
about  an  average  constant  strength.  Many  electrical  determina- 
tions require  for  their  accuracy  either  that  the  horizontal  intensity 
should  remain  constant  or  that  its  fluctuations  should  be  taken 
into  account.  As  the  latter  is  the  only  alternative,  a  means  must 
be  had  of  determining,  at  any  moment,  whether  the  intensity  has 
changed,  and,  if  so,  how  much. 

One  method  is  to  employ  a  magnetometer  (Fig.  342),  which  is 
rendered  nearly  astatic  by  a  supplementary  bar  magnet.  (A  needle 
is  astatic  when  the  earth  has  no  directive  effect  upon  it.)  This 
auxiliary  magnet  is  placed  north  and  south,  directly  under,  or 
over,  the  needle  of  the  magnetometer.  When  placed  at  a  proper 
distance  above  the  needle,  depending  upon  its  strength,  it  will  act 
upon  the  needle  with  the  same  force  as  the  earth,  only  in  an  op- 
posite direction.  It  will  thus  neutralize  the  influence  of  the  earth 
and  the  needle  can  turn  into  any  position.  If  the  magnet  be 
brought  a  little  nearer,  the  needle  will  suddenly  turn  around  and 
its  north-seeking  pole  will  point  south.  Now,  by  a  little  delicate 
manipulation,  the  needle  may  be  made  to  point  nearly  east  and 
west.  In  this  position  it  is  very  sensitive.  Any  small  increase  in 
the  earth's  intensity  will  cause  its  north-seeking  end  to  turn  to  the 
north,  and  any  decrease  to  the  south.  Thus,  by  looking  through 
the  telescope  at  the  mirror,  any  change  in 
the  intensity  can  be  detected  at  any  mo-  FIG.  347. 

ment,  and  the  amount  of  change  can  be 
arrived  at  by  calculation. 

Astatic  needles  are  of  great  value  in 
electrical  measurements.  Liberated  from 
the  earth's  directive  action  they  may  still 
be  affected  by  electrical  currents.  Another 
method  of  obtaining  this  end  is  shown  in 
Fig.  347. 

A  compound  needle,  consisting  of  two 
simple  needles  fixed  upon  a  wire,  with 

their  unlike  poles  opposed,  may  be  suspended  in  any  of  the  usual 
modes.  If  the  needles  are  exactly  equal  in  all  respects  the  system 


394  ELECTRICITY     AND     MAGNETISM. 

will  be  perfectly  astatic.     The  condition  of  perfect  equality  in  all 
the  conditions  is  never  realized. 

626.  Magnetic  Charts. — These  are  maps  of  a  country,  or  of 
the  world,  on  which  are  laid  down  the  systems  of  curves  which 
have  been  described.     But  for  the  use  of  the  navigator,  only  the 
isogonic  lines,  or  lines  of  equal  declination,  are  essential.     There 
are  large  portions  of  the  globe  which  have  as  yet  been  too  imper- 
fectly examined  for  the  several  systems  of  curves  to  be  accurately 
mapped..    It  must  be  remembered,  too,  that  the  earth  is  slowly 
but  constantly  undergoing  magnetic  changes,  by  which,  at  any 
given  place,  the  declination,  dip,  and  intensity  are  all  essentially 
altered  after  the  lapse  of  years.     A  chart,  therefore,  which  would 
be  accurate  for  the  middle  of  the  nineteenth  century,  will  be,  to 
some  extent,  incorrect  at  its  close. 

627.  The    Declination    Compass. — This   instrument   con- 
sists of  a  magnetic  needle  suspended  in  the  centre  of  a  cylindrical 
brass  box  covered  with  glass ;  on  the  bottom  of  the  box  within  i» 
fastened  a  circular  card,  divided  into  degrees  and  minutes,  from 
0°  to  90°  on  the  several  quadrants.     On  the  top  of  the  box  are  two 
uprights,  either  for  holding  sight-lines  or  for  supporting  a  small 
telescope,  by  which  directions  are  fixed.     The  quadrants  on  the 
card  in  the  box  are  graduated  from  that  diameter  which  is  verti- 
cally beneath  the  line  of  sight. 

When  the  axis  of  vision  is  directed  along  a  given  line,  the 
needle  shows  how  many  degrees  that  line  is  inclined  to  the  mag- 
netic meridian.  In  order  that  the  angle  between  the  line  and  the 
geographical  meridian  may  be  found,  the  declination  of  the  needle 
for  the  place  must  be  known. 

628.  The  Mariner's  Compass. — In  the  mariner's  compass 
(Fig.  348)  the  card  is  made  as  light  as  possible,  and  attached  to 

the  needle,  so  that  the  north  and 
south  points  marked  on  the  card 
always  coincide  with  the  magnetic 
meridian.  The  index,  by  which 
the  direction  of  the  ship  is  read, 
consists  of  a  pair  of  vertical  lines, 
diametrically  opposite  to  each 
other,  on  the  interior  of  the  box. 
These  lines,  one  of  which  is  seen 
at  a,  are  in  the  plane  of  the  ship's 
keel.  Hence,  the  degree  of  the  card  which  is  against  either  of 
the  lines  shows  at  once  both  the  angle  with  the  magnetic  meridian 
and  the  quadrant  in  which  that  angle  lies. 


AURORA     BOREALIS.  395 

In  order  that  the  top  of  the  box  may  always  be  in  a  horizontal 
position,  and  the  needle  as  free  as  possible  from  agitation  by  the 
rolling  of  the  ship,  the  box,  B,  is  suspended  in  gimbals.  The 
pivots,  A,  A,  on  opposite  sides  of  the  box,  are  centred  in  the  brass 
ring,  G,  D,  while  this  ring  rests  on  an  axis,  which  has  its  bear- 
ings in  the  supports,  E,  E.  These  two  axes  are  at  right  angles  to 
each  other,  and  intersect  at  the  point  where  the  needle  rests  on  its 
pivot.  Therefore,  whatever  position  the  supports,  E,  E,  may  have, 
the  box,  having  its  principal  weight  in  the  lower  part,  maintains 
its  upright  position,  and  the  centre  of  the  needle  is  not  moved  by 
the  revolutions  on  the  two  axes. 

On  account  of  the  dip,  which  increases  with  the  distance  from 
the  equator,  and  is  reversed  by  going  from  one  hemisphere  to  the 
other,  the  needle  needs  to  be  loaded  by  a  small  adjustable  weight, 
if  it  is  to  be  used  in  extensive  voyages  to  the  north  or  south. 

629.  Aurora  Borealis. — This  phenomenon  is  usually  accom- 
panied by  a  disturbance  of  the  needle,  thus  affording  visible  indi- 
cations of  a  magnetic  storm ;   but  the  contrary  is  by  no  means 
generally  true,  that  a  magnetic  storm  is  accompanied  by  auroral 
light.     The  connection  of  the  aurora  borealis  with  magnetism  is 
manifested  not  only  by  the  disturbance  of  the  needle,  but  also  by 
the  fact  that  the  streamers  are  parallel  to  the  dipping-needle,  as 
is  proved  by  their  apparent  convergence  to  that  point  of  the  sky 
to  which  the  dipping-needle  is  directed.     This  convergence  is  the 
effect  of  perspective,  the  lines  being  in  fact  straight  and  parallel. 

630.  Why  is  the  Earth  a  Magnet? — Modern  discoveries 
in  electro-magnetism  and  thermo-electricity  furnish  a  clew  to  the 
hypothesis  which  generally  prevails  at  this  day.     Attention  has 
been  drawn  to  the  remarkable  agreement  between  the  isothermal 
and  the  isomagnetic  lines  of  the  globe.     The  former  descend  in 
crossing  the  Atlantic  Ocean  toward  America,  and  there  are  two* 
poles  of  maximum  cold  in  the  northern  hemisphere.     The  isoclinic 
and  the  isodynamic  curves  also  descend  to  lower  latitudes  in  cross- 
ing the  Atlantic  westward  ;  so  that,  at  a  given  latitude,  the  degree 
of  cold,  the  magnetic  dip,  and  the  magnetic  intensity,  are  each  con- 
siderably greater  on  the  American  than  on  the  Eui'opean  coast. 
This  is  only  an  instance  of  the  general  correspondence  between 
these  different  systems  of  curves.     It  has  likewise  been  noticed 
(Art.  619)  that  the  needle  has  a  movement  diurnally,  varying  west- 
ward during  the  middle  of  the  day,  and  eastward  at  evening,  and 
that  this  oscillation  is  generally  much  greater  in  the  hot  season 
than  the  cold.     It  is  obvious,  therefore,  that  the  development  of 
magnetism  in  the  earth  is  intimately  connected  with  the  tempera- 


396  ELECTRICITY     AND     MAGNETISM. 

ture  of  its  surface.  Hence  it  has  been  supposed  that  the  heat 
received  from  the  sun  excites  electric  currents  in  the  materials  of 
the  earth's  surface,  and  these  give  rise  to  the  magnetic  phenomena. 
Most  interesting  is  the  hypothesis  recently  projected  by  Pro- 
fessor Bigelow,  viz.,  the  earth  is  revolving  and  moving  in  a 
magnetic  field,  which  is  created  by  the  sun.  According  to  this 
the  earth  is  a  magnet  by  induction  and  the  variations  in  its  mag- 
netism are  caused  by  differences  in  the  strength  of  the  field 
through  which  it  is  moving. 


CHAPTER    V. 

CURRENT     ELECTRICITY. 

631.  Electricity  in  Motion. — It  has  been   seen  (Art.  571) 
that  when  conductors  which  have  a  difference  of  electrical  poten- 
tial are  connected  together  by  a  conducting  substance,  a  flow,  or 
current,   of  electricity  from  the  higher  to  lower  potential    takes 
place.     This  current,  however,  lasts  for  an  instant  only,  and  any 
phenomena  due  directly  to  the  flow  would  have  to  be  observed 
during  that  instant.     If  by  any  means  the  difference  of  potential 
of  the  bodies  could  be  maintained  in  spite  of  their  being  con- 
nected, a  continuous  curi'ent  would  be  made  to  traverse  the  con- 
necting conductor.      Such  a  means  was  accidentally  discovered  in 
1786,  by  Galvani,  Professor  of  Anatomy  at  Bologna.     After  exper- 
imenting, one  day,   upon  the  effects  of  statical  electricity  on  a 
frog's  leg,  he  hung  the  moist  leg,  by  means  of  a  copper  hook,  upon 
an  iron  window-guard.     He  then  noticed  that,  whenever  the  free 
end  of  the  leg  touched  the  guard,  it  gave  a  spasmodic  twitch,   as 
though  a  statical  charge  had  been  passed  through  it.     He  accord- 
ingly surmised  that  he  had  found  a  new  method  of  obtaining 
electricity. 

632.  Galvanic    Cells.  —  Galvani's   discovery   has  developed 
into  the    Galvanic  Cell  or  Element — an  arrangement  of  apparatus 
designed  to  give  a  continuous  flow  of  electricity. 

If,  when  two  different  substances  are  submerged  in  an  oxidizing 
fluid,*  one  of  them  has  a  greater  affinity  for  oxygen  than  the  other, 
then  a  difference  of  potential  will  be  set  up  between  them.  The 
one  having  the  greater  affinity  will  have  a  lower  potential.  If  de- 
sirable, the  substances  may  be  considered  as  having  become  elec- 
trified— the  least  oxidizable  positively,  and  the  other  negatively. 

*  For  simplicity,  affinity  for  oxygen  is  alone  mentioned  here.  The  princi- 
ple is  true  for  any  chemical  affinity. 


ELECTROMOTIVE     FORCE.  397 

\ 

If,  now,  the  substances  be  connected  by  a  wire,  a  current  will 
flow  through  it  from  the  higher  to  the  lower  potential.  As  long  as 
the  chemical  action  keeps  up,  the  difference  of  potential  and  the 
current  resulting  from  it  will  be  maintained. 

If,  for  example,  the  two  substances  were  copper  and  zinc 
plates,  and  the  fluid  was  dilute  sulphuric 
acid  (Fig.  349),  the  zinc,  having  greater 
affinity  for  the  oxygen  of  the  acid,  would 
have  a  lower  potential  than  the  cop- 
per. Upon  connecting  them  by  a  wire,  a 
cui'rent  would  flow  from  the  copper  to  the 
zinc.  This  would  be  a  simple  galvanic  ele- 
ment. 

The  arrangement  need  not  be  as  shown 
in  the  figure,  for  a  zinc  rod,  wrapped  in 
blotting-paper,  upon  which  is  wound  bare 
-copper  wire,  would,  upon  moistening  the  paper  with  dilute  acid, 
give  a  current. 

The  flow  from  copper  to  zinc,  in  the  connecting  wire,  is  always 
ticcompanied  by  an  equal  flow  from  zinc  to  copper  through  the 
submerging  fluid.  (This  latter  flow  is  found  necessary  for  the 
maintenance  of  the  potential  difference.)  Thus,  if  we  start  at  any 
point  and  follow  the  current,  we  will  eventually  come  back  to  the 
point  whence  we  started,  i.e.,  a  current  of  electricity  always  flows  in 
a  closed  circuit. 

633.  Electromotive  Force. — The  difference  of  potential  set 
up  in  a  galvanic  element  is  clue  to  an  Electromotive  Force,  which 
is  generally  represented  by  the  letters  E.  M.  F.  Its  amount  de- 
pends upon  the  nature  of  the  two  substances  employed — their 
relative  affinities  for  the  active  part  of  the  fluid.  In  dilute  sul- 
phuric acid  they  arrange  themselves  in  the  following  order : 

Hydrogen, 

Zinc, 

Iron, 

Lead, 

Nickel, 

Bismuth, 

Copper, 

Carbon, 

Silver, 

Platinum, 

Oxygen. 

Of  the  metals  given,  zinc  has  the  greatest  affinity  for  oxygen, 
and  platinum  the  least.  These  two  metals  then  would  give  the 


398  ELECTRICITY     AND     MAGNETISM. 

greatest  E.  M.  F.  Platinum  and  silver  would  give  hardly  any.  If 
two  elements  be  constructed,  using  lead-zinc  for  one  and  lead-cop- 
per for  the  other,  the  current  would  flow  out  of  the  lead  in  the 
first  case,  and  into  the  lead  in  the  second. 

The  absolute  electrostatic  unit  of  potential  difference  is  too 
large  for  practical  purposes,  hence  a  practical  unit  of  E.  M.  F., 
called  the  volt  (=  ?J7  electrostatic  unit)  is  employed.  The  E.  M. 
F.  of  copper-zinc  in  dilute  sulphuric  acid,  at  the  instant  of  making 
first  contact,  is  0.921  volt. 

The  E.  M.  F.  of  a  cell  is  independent  of  the  size  of  the  electrodes. 

A  copper-zinc  cell  of  1  sq.  cm.  electrodes  has  the  same  E.  M. 
F.  as  one  with  1,000  sq.  cms. 

The  total  E.  M.  F.  in  a  circuit  is  equal  to  the  algebraic  sum  of 
the  separate  E.  M.  F.'s. 

Thus,  if  two  copper-zinc  cells  be  connected  in  a  circuit  in  the 
order  (Cu— Zu)  — (Zn  — Cu)  one  will  tend  to  send  a  current  in  one 
direction,  and  the  other  in  the  opposite  direction.  The  result  will 
be  no  current  at  all. 

If,  in  trying  this  experiment,  one  of  the  cells  be  very  large  and 
the  other  very  small,  the  fact  that  no  current  flows  would  illustrate 
the  fact  that  the  E.  M.  F.  is  independent  of  the  size  of  electrodes. 

634.  Polarization. — If  a  copper-zinc  sulphuric-acid  cell  be 
connected  with  an  electric  bell  (or  any  other  current  indicator)  it 
will  at  first  ring  loudly,  but  will  soon  weaken,  and  finally  cease  to 
give  a  sound.     Upon  investigating  the  cause  of  this  weakening  it 
will  be  found  that  the  E.  M.  F.  has  fallen  from  1  volt  to  possibly 
0.2  volt.     This  is  because  the  current,  which  the  element  has  sent 
through  its  own  liquid,  has  decomposed  that  liquid,  and  hydrogen 
(Art.  678)  has  been  deposited  upon  the  copper  and  oxygen  upon 
the  zinc.     The  oxygen   immediately   enters  into   chemical  union 
with  the  zinc,  but  the  hydrogen  remains  in  its  gaseous  form.     The 
hydrogen,  from  its  affinity  for  oxygen,  sets  up  a  counter  E.  M.  F., 
tending  to  send  a  current  in  an  opposite  direction.     The  resulting 
current  is  smaller  than  at  first,  and  the  cell  is  said  to  have  become 
polarized. 

635.  Types  of  Batteries.— (A  collection  of  galvanic  cells  is- 
termed  a  battery.}     Practical  cells,  designed  for  giving  a  constant 
flow  of   electricity,    employ   different   methods   for   avoiding   the 
counter   E.  M.  F.  of   polarization.     The  market  affords  a  great 
variety,  but  we  need  consider  but  three  : 

BUNSEN'S  CELL. — As  has  been  shown,  the  counter  E.  M.  F.  is. 
due  to  hydrogen  upon  the  electrode  having  the  higher  potential. 


BATTERIES. 


399 


FIG.  350. 


In  Bunsen's  cell  this  hydrogen  is  made  to  combine  with  oxygen 
furnished  by  nitric  acid.  The  cell  (Fig.  350)  employs  two 
different  acids,  which  are  kept  separate 
by  a  porous,  unglazed  cup.  This  allows 
the  electricity  to  flow,  but  prevents  a 
free  mixture  of  the  acids.  Outside  the 
cup  is  zinc  in  dilute  sulphuric  acid  ;  in- 
side is  carbon  in  nitric  acid.  The  hy- 
drogen, which  above  was  deposited  upon 
the  copper,  now  comes  off  at  the  carbon. 
Instead  of  being  allowed  to  exert  a  coun- 
ter E.  M.  F.,  it  is  immediately  oxidized 
by  the  nitric  acid.  On  the  other  hand, 
this  acid  is  prevented  by  the  porous  cup 
from  violently  attacking  the  zinc. 

This  cell  has  an  E.  M.  F.  of  1.8  volt, 
and  is  capable  of  maintaining  it  for  a 
long  time.  It  is  a  disagreeable  cell  to 
work  with  because  of  the  nitric-acid 
fumes.  These  fumes  can  be  avoided  by 
substituting  a  solution  of  bichromate  of 
potash  for  the  nitric  acid.  It  is  also  an  active  oxidizer ;  but,  in 
time,  large  crystals  form  inside  the  walls  of  the  porous  cup  and 
cause  them  to  break  in  pieces, 

DAXIELL'S  CELL. — This  cell  is  more  uqpd'than  anv  other,  in  the 
laboratory.  It  also  employs  two  liquids  and 
a  porous  cup.  The  arrangement  is  (Fig. 
351)  zinc  in  dilute  sulphuric  acid  inside  the 
cup,  and  copper  in  copper  sulphate  outside. 
The  cell's  own  current,  instead  of  depositing 
hydrogen  upon  the  copper,  deposits  copper 
from  its  sulphate.  Now  copper  upon  copper 
cannot  alter  the  E.  M.  F.  of  the  cell,  and 
hence  the  Daniell  has  the  most  constant  E.  M. 
F.  of  ordinary  cells.  The  E.  M.  F.  depends 
somewhat  upon  the  dilution  of  the  sul- 
phuric acid,  but  is  very  nearly  1  volt  for  any 
arrangement. 

A  modified  form  of  Daniell's  cell,  called  the  gravity  cell,  is  used 
in  telegraphy.  The  porous  cup  is  dispensed  with,  and  the  two 
liquids  are  kept  separate  by  the  action  of  gravity.  The  dilute  sul- 
phuric acid  is  floated  on  top  of  the  copper  sulphate. 

LECLANCHE  CELL. — There  are  more  of  this  form  of  cell  in  use 
than  of  all  others  put  together.  They  are  not  designed  to  main- 


FIG.  351. 


400 


ELECTRICITY     AND     MAGNETISM. 


tain  a  constant  E.  M.  F.  for  any  great  length  of  time.    'They  are 
intended  for  purposes  where  a  current  is  needed  for  only  a  few 
moments   at   most,   as   for   electric 
FIG.  352.  bells  or  on  telephone  circuits.    After 

use  they  regain  their  original  E,  M. 
F.  The  arrangement  (Fig.  352)  is 
zinc  and  carbon  in  a  solution  of  sal 
ammoniac  (NH4C1).  An  attempt  is 
made  to  get  rid  of  the  hydrogen  of 
polarization  by  surrounding  or  mix- 
ing the  carbon  with  an  oxide  of 
manganese.  This  eventually  oxi- 
dizes the  hydrogen,  but  not  as 
rapidly  as  the  nitric  acid  in  Bun- 
sen's  cell.  Some  forms  of  Le- 
clanche  cell  employ  a  porous  cup 
containing  the  carbon  amid  the 
manganese  oxide.  The  E.  M.  F.  of 
a  fresh  Leclanche  is  1.5  volt. 


636.  Combustion  of  Zinc. — 
Nearly  all  batteries  employ  zincs  for 
the  lower  potential  electrode.  A 
current  flow  is  always  accompanied 

by  an  oxidation  of  zinc,  and  the  energy  which  accompanies  the 
current  comes  from  this  oxidation.  This  is  parallel  to  the  case  of 
a  steam-engine,  where  the  energy  comes  from  the  oxidation  of  the 
fuel  under  the  boiler. 


637.  Amalgamation  of  Zincs. — Ordinary  commercial  zinc  is 
impure.  If  this  impurity  were,  say  copper,  and  a  particle  should 
be  embedded  near  the  surface  of  a  zinc  electrode,  then,  upon  im- 
mersing in  acid,  the  zinc  and  copper  would  form  a  small  cell  by 
themselves.  This  would  be  giving  a  current,  whether  the  complete 
cell  were  in  use  or  not,  and  would  be  continually  wasting  zinc. 
This  wasting  of  zinc,  because  of  impurities  in  it,  is  called  local 
action  of  the  cell. 

It  has  been  found  that  local  action  can  be  prevented  by  amal- 
gamating the  zinc.  This  is  done  by  dipping  the  zinc  in  acid  and 
then  in  mercury.  The  mercury  unites  with  the  zinc  and  floats 
the  impurities  to  its  surface.  These  are  then  detached  by  the  gas 
bubbles,  which  are  caused  by  their  union  with  the  acid.  The  zinc 
of  the  amalgam  is  oxidized  by  the  action  of  the  battery,  but  the 
mercury  remains  unaffected.  It  is,  therefore,  constantly  going- 


RESISTANCE.  401 

into  combination  with  new  zinc,  as  the  action  of  the  battery  con- 
tinues. 

638.  Practical  Units  of  Current  and  Quantity.  —  As,  in 
considering  the  flow  of  water  in  a  pipe,  we  give  the  current  a 
definite  value  of  say  so  many  gallons  per  hour,  so  we  can  give   a 
definite  value  to  the  electrical  current. 

The  quantity  of  water  passing  through  any  cross-section  of  a 
water-pipe  of  varying  diameter  is  the  same  for  the  same  time  and 
current.  Likewise 

The  quantity  of  electricity  passing  in  the  unit  time  through  any 
cross-section  of  a  simple  undivided  circuit  is  the  same  for  the  same 
current. 

To  obtain  a  unit  for  current  we  have  only  to  use  the  unit  for  quan- 
tity and  the  one  for  time.  Now,  the  absolute  electrostatic  unit  of 
quantity  is  not  of  convenient  size  for  practical  purposes.  Hence,  a 
new  unit,  termed  the  Coulomb,  is  employed.  It  equals  3,000,000,000 
absolute  electrostatic  units.  We  have,  then, 

The  practical  unit  of  current,  the  ampere,  is  that  current  which 
delivers  one  coulomb  per  second  to  any  cross-section  of  the  circuit. 

639.  Resistance.  —  All  substance  offers  a  resistance  to  the  flow 
of  electricity.     Just  as  motion  against  resisting  friction  produces 
heat,  so  a  current  overcoming  electrical  resistance  produces  heat. 

The  resistance  offered  by  a  given  conductor  depends  upon  two 
things,  viz.,  the  character  of  the  substance  and  its  shape.  If  we 
represent  the  length  of  a  conductor  by  I  metres,  its  cross-section 
by  q  sq.  mm.,  then  its  electrical  resistance 


s  being  a  constant  depending  upon  the  character  of  the  substance 
and  termed  its  specific  resistance.*  If  we  assumed  s,  I,  and  q  each 
equal  to  unity,  we  would  have  a  unit  resistance.  A  unit,  much 
used  in  Germany,  the  Siemen's  quicksilver  unit,  is  defined  by  as- 
suming that  s  for  quicksilver  ut  0°  C.  is  unity.  Hence,  Siemens  unit 
of  resistance  is  the  resistance  offered  by  a  column  of  quicksilver  1  me- 
tre long  and  1  sq.  mm.  cross-section,  at  0°  C. 

The  international  practical  unit,  the  legal  ohm,  is  a  little  larger 
than  the  Siemen's  unit. 

1  ohm  =  1.06  Siemen's  unit. 

If  R  represents  the  resistance  of  a  conductor,  '/«  evidently  repre- 
sents its  conductivity—  the  greater  the  resistance  the  smaller  the 

*  The  absolute  Hperific  resistances  depending  upon  I  and  q  being  measured  in 
centimetres,  and  7?  in  absolute  units,  are  expressed  by  unwieldly  numbers,  and 
a  comprehension  of  the  subject  does  not  require  their  consideration. 


402  ELECTRICITY     AND     MAGNETISM. 

conducting  power.     Accordingly,  7s  can  be  called  the  specific  con- 
ductivity of  a  substance. 

SPECIFIC  CONDUCTIVITIES,  k  =  l/s. 
Mercury  .  .  .  .  .............  ...................     1.06 

Silver  ......................................  63. 

Copper  .....................................  58. 

Iron  .......................................     7.4  to  9.5 

Platinum  ...................................     6.9 

German  silver  ..............................     2.5  to  6.4 

.  Zn  S04  (sat.  sol.)  .........................  ...       .0000043 

Pure  Water  ..................................  000000000025 

Glass  ........................................  0 

These  figures  evidently  represent  the  length  in  metres  of  a 
wire  of  1  sq.  mm.  cross-section,  that  the  resistance  may  be  1  ohm. 

Their  application  can  be  best  understood  by  an  example.  Deter- 
mine the  resistance  of  a  copper  wire  11.6  m.  long  and  0.1  sq.  mm. 
in  cross-section. 


Silver  and  copper  are  the  best  conductors  we  have.  Because  of 
the  expense  of  the  former,  copper  is  universally  employed  on 
electrical  circuits.  In  fact,  some  modern  copper  is  said  to  con- 
duct better  than  silver.  Absolutely  pure  water  is  probably  a  non- 
conductor. The  purest  water  yet  obtained,  if  placed  in  a  tube 
of  unit  diameter  and  1  mm.  long  would  offer  the  same  resistance 
as  a  copper  wire  of  same  diameter,  but  as  long  as  the  orbit  of  the 
moon. 

The  influence  of  specific  conductivity  upon  resistance  can  be 
prettily  shown  by  the  following  experiment  :  Pass  the  current 
from  a  dynamo  through  an  electric  lamp,  and,  by  means  of  two 
electrodes,  through  a  vessel  of  rain-water.  As  long  as  the  water  is 
pure  the  lamp  will  not  be  illuminated.  Place  a  few  drops  of  sul- 
phuric acid  in  the  water,  and  the  lamp  will  instantly  commence  to 
glow. 

640.  Influence  of  Temperature.  —  The  resistance  of  con- 
ductors changes  with  the  temperature.  In  all  metals  an  increase 
of  temperature  increases  the  resistance.  At  ordinary  temperatures 
the  increase  for  most  pure  metals  is  0.004  of  the  whole,  per  de- 
gree centigrade.  (  The  amount  for  German  silver  is  about  0.0003. 

Carbon  and  liquids  decrease  in  resistance  when  the  temperature 
is  raised.  The  change  per  degree  for  liquids  is  between  two  and 
three  per  cent. 

The  dependence  of  resistance  upon  temperature  furnishes  a 
means  of  measuring  the  latter.  A  conductor  of  large  temperature 


OHM'S    LAW.  403 

coefficient  is  subjected  to  the  heat  whose  temperature  is  to  be  de- 
termined, and  while  still  in  place  its  resistance  is  measured.  The 
increase  of  resistance  furnishes  data  for  calculation  of  the  tempera- 
ture. By  this  means  Professor  Langley  has  measured  the  heat 
radiated  from  the  moon. 


641.  Ohm's    Law. — The  three  electrical   magnitudes — cur- 
rent, E.  M.  F.,  and  resistance — are  connected  together  by  an  impor- 
tant relation  called  Ohm's  law.     Letting  E  represent,  in  volts,  the 
algebraic  sum  of  all  the  E.  M.  F.'s  of  a  circuit,  R  the  sum  of  all 
the  resistances,  in  ohms  (of  battery,  conducting  wires,   and  all 
instruments  in  circuit),  then  this  law  states,  the  current  strength 
in  amperes, 

The  strength  of  the  current  varies  directly  as  tAe  E.  M.  F.,  and 
inversely  as  the  resistance. 

642.  Divided   Circuits. — Shunts.  —  If  a  current  from  a 
battery  of   E.  M.  F.   =  E  and   internal  resistance  —  r  be  sent 
through  a  wire  which  divides  at  a  certain  point  (Fig.  353)  into 
two   branches,   which   however   re- 
unite further  on,  and  if  the  resist- 
ances of  the  undivided  conductor 

and  its  branches  are  R,  1\,  i\  re- 
spectively, then  the  substitution  of 
r  +  R  -\-  TI  +  rz  in  Ohm's  law 
would  not  give  the  correct  current 
strength.  The  reason  for  this  is 
that  the  whole  current  does  not 
pass  through  each  of  the  branches 
7-j  and  r2.  They  each  take  a  por- 
tion of  the  current,  depending  upon  their  resistances.  A  single 
conductor  might  be  found,  which,  if  substituted  for  the  two, 
would  leave  the  current  in  R  unchanged.  The  resistance  of  this 
single  conductor  might  be  called  the  equivalent  resistance  of  the 
branches.  To  determine  this  equivalent  resistance  it  is  most  con- 
venient to  consider  the  conductivities  of  the  branches  Evidently 
the  conductivity  of  the  single  replacing  conductor  must  equal  the 
sum  of  the  conductivities  of  the  separate  paths.  But  the  con- 
ductivities are  the  reciprocals  of  the  resistances.  Hence  we  have 


404  ELECTRICITY     AND     MAGNETISM. 

The   current   C  flowing   through   the   undivided   portion   of  the 
circuit,  e.g.,  through  R,  would  be,  by  Ohm's  law, 
E      __ E__ 

I  R  +  r  -h  R'       R  +  r  +  ~^^- 

r,  +  r, 

In  the  same  manner  the  equivalent  resistance  of  any  number  of  dif- 
ferent paths  may  be  determined. 

When  a  conductor  is  placed  so  as  to  take  a  portion  of  the 
current  which  is  passing  through  another  conductor  it  is  called 
a  shunt  and  the  current  is  said  to  be  shunted. 

Many  delicate  instruments  for  measuring  electrical  quantities 
would  be  ruined  if  the  whole  current  passed  through  them.  In 
such  cases  a  portion  of  the  current  is  shunted  off  from  the  instru- 
ment. From  the  known  resistances  of  the  instrument  and  the 
shunt  the  quantities  to  be  determined  can  be  calculated. 

643.  Ratio  of  Currents  in  Shunts. — In  order  to  determine 
the  portion  of  the  current  flowing  in  any  branch  of  a  divided 
circuit,   we  must  consider  that  the  whole  current  is  carried  by 
the  branches  as  a  whole.     Letting  C  —  current  in  undivided  por- 
tion (Fig.  353)  and  c,  and  c2  =  currents  in  r,  and  ra,  we  have 

C  =  Cl  +  ca. 

Again,  bearing  in  mind  that  the  difference  of  potential  (E.  M.  F.) 
between  th'e  ends  of  each  branch  is  the  same  =  e,  we  have,  by 
Ohm's  law, 

e  .  e 

c,  =  —  >  c,  =  — ,  etc. 
r,  ra 

.'.  c,  :  c,  :  etc.  =  —  :  —  :  etc. 
r,       r2 

The  currents  carried  by  different  branches  between  two  points  of  a 
circuit  are  inversely  as  the  resistances  of  the  branches. 

644.  Fall   of  Potential. — If  we  have  a  battery  connected 
through  a  uniform  straight  wire  of  given  resistance,  we  may  con- 
sider the  potential  at  the  zinc  end  to  be  =  zero,  and  that  at  the 
other  end  positive,  and  =  E.  M.  F.  of  the  battery.     Now,  inasmuch 
as  the  resistances  of  equal  lengths  of  the  wire  are  the  same,  the 
potential  at  the  middle  of  the  wire  equals  one-half  the  E.  M.  F. 
At  one-quarter  the  distance  from  each  end  of  the  wire  the  poten- 
tials are  one-quarter  and   three-quarters   of   the  E.  M.  F.     The 
potential  varies  all  along  the  wire,  from  zero  at  the  zinc  end  to  E. 
M.  F.  at  the  other  end.     If  we  commence  at  the  other  end  we  may 
say  that 

The  potential  falls  directly  as  the  resistance. 

Of  course,  if  the  conductor  were  not  homogeneous,  e.g.,  made 


WHEATSTONE'S    BRIDGE. 


405 


of  copper  and  then  German  silver,  the  fall  would  not  be  the  same 
for  the  same  lengths.     It  would  be  more  rapid  in  the  German 
silver  portion    of    the    circuit 
than  in  the  copper  portion.  FIG.  354. 

645.  Resistance  Boxes 
or  Rheostats.  —  These  are 
boxes  (Fig.  354)  containing  dif- 
ferent spools  of  wire,  whose 
resistances  have  been  deter- 
mined, and  which  can  be  used 
for  standards  of  comparison. 
German  silver  wire  is  gener- 
ally used,  because  its  resist- 
ance changes  least  with  the 
temperature.  The  proper  length 
of  wire  is  taken,  and,  after  be- 
ing doubled  at  its  middle  (Fig.  355),  is  wound  upon  a  spool,  the 
two  parts  being  wound  side  by  side.  This  is  indicated  in  the 
figure.  The  reason  for  this  doubling  is 
to  avoid  the  self-induction  disturbances 
of  the  wire  (Art.  668).  The  spool,  when 
wound,  is  placed  inside  the  box  and  the 
terminals  are  fastened  to  two  separate 
brass  blocks  on  the  top  of  the  box.  To 
each  of  these  blocks  is  fastened  one  end 
of  the  two  neighboring  coils.  In  the 
_A  figure,  a  and  6  of  one  resistance  are 

fastened  to  blocks  E  and  H.     The  ends 

c  and  d  of  the  neighboring  coil  are  fastened,  one  to  H  and  the 
other  to  M.  The  blocks  can  be  connected  together  at  will  by 
brass  plugs  fitting  into  holes  between  them. 

Suppose,  now,  that  the  two  terminals  of  a  battery  be  connected 
with  E  and  M  respectively.  If  the  plugs  1  and  2  be  removed,  the 
current  will  be  obliged  to  traverse  both  of  the  resistance  coils.  If 
plug  1  be  inserted,  the  current  will  divide  between  the  plug  and 
the  coil.  But  the  resistance  of  the  plug  is  infinitesimal,  and  hence, 
practically,  the  whole  current  passes  through  it  and  none  through 
the  coil. 

When  a  box  of  coils  is  inserted  in  a  circuit,  the  resistance  can 
be  varied  at  will,  by  simply  inserting1  or  pulling  out  plugs. 


FIG.  355. 


646.  Wheatstone's  Bridge. — An  important  application  of 
the  preceding  principles  is  Wheatstone's  bridge.     It  is  an  arrange- 


406 


ELECTRICITY     AND     MAGNETISM. 


ment  of  apparatus  by  means  of  which  resistances  can  be  very  accu- 
rately determined. 

If  a  current  of  electricity  arriving  at  F  (Fig.  356)  -divides  and 


by  two  paths,  A  B  and  CD,  to  V,  where  the  paths- 
unite,  then  the  difference  of  potential  (V-V)  is  acting  on  both 
paths.  The  potential  along  each  path  must  fall  from  V  to  V.  If 
at  any  point  of  the  path  A  B,  the  potential  is  g,  then  some  point  (g') 
of  the  other  path,  C  D,  can  be  found  having  the  same  potential. 
If  these  two  points,  g  and  g'  be  connected  through  a  galvanometer 
or  any  other  current  detector,  no  current  will  flow,  because  there  is 
no  potential  difference  between  g  and  g'. 

Inasmuch  as  the  fall  or  loss  of  potential  along  each  path  is  pro- 
portional to  the  resistance,  the  loss  in  passing  A  must  be  the  same 
as  in  passing  C,  and  the  resistance  of  A  must  bear  the  same  ratio 
to  A  +  B  as  C  does  to  C  +  D.  In  order  that  the  potentials  may  be 
the  same  at  g  and  g',  it  must  be  true  that  the  resistances  follow 
the  proportion 

A-.E  =  C:D. 

In  Wheatstone's  bridge,  when  no  current  passes  through  the 
galvanometer,  the  products  of  the  opposite  resistances  are  equal. 

The  method  of  determining  resistances  by  the  bridge  is  to  place 
the  unknown  resistance  in  one  arm  of  the  bridge,  as  D.  Known 
resistances  are  placed  in  A  and  B,  and  a  resistance-box  in  G.  By 
manipulating  the  plugs  in  C  a  balance  can  be  made  so  that  no  cur- 
rent flows  through  the  galvanometer.  A,  B,  and  G  are  known,  and 
the  required  resistance 


It  is  sometimes  convenient  to  make  G  constant,  and  vary  bo^h 
A  and  B  until  a  balance  is  obtained,  A  and  B  consisting  of  parts 
of  the  same  straight  wire  of  uniform  diameter.  The  balance  is 


ARRANGEMENT     OF     CELLS.  407 

obtained  by  sliding  the  contact  (g)  with  the  galvanometer  along 
this  wire.  The  resistances  of  A  and  B  are  then  proportional  to 

their  lengths. 

647.  Cells  in  Series  and  in  Multiple  Arc.  —  A  battery  of 
two  cells  can  be  connected  to  a  circuit 
in  two    different  manners.     The  copper  FIG.  357. 

of  one  may  be  connected  to  the  zinc  of  1  1    II 

the  other  (Fig.  357),  and  the  circuit  4  -  \\{\  - 
connected  to  the  remaining  zinc  and  '  [  '] 

copper.     The   cells   would    then    be   in 

series.     Again,  the  coppers  of  each  and  the  zincs  of  each  might 

be  connected  together  and  the  circuit  connected  to  these  short 

connecting  wires  (Fig.  358).     The  two 

cells  are  then  said  -to  be  in  multiple  arc. 

\\  Let   us   consider   the   results   of    these 

/||\  different  arrangements.     Eepresent  the 

<          f         j  -     E.  M.  F.  of  each  cell  by  E,  and  the  in- 

VJL/  ternal  resistance  by  r. 

Evidently,  when  in  series,  the  E.  M. 
F.  of  the  circuit  is  equal  to  the  sum  of 

the  two  E's,  and  the  internal  resistance  of  the  battery  is  2  r.  If 
the  resistance  of  the  total  external  circuit  be  B,  then  the  current, 
when  the  cells  are  in  series, 


~  R  +  2r 

When  the  cells  are  in  multiple  arc,  the  E.  M.  F.  is  no  greater 
than  for  a  single  cell.  The  two  cells  are  like  a  single  cell  of  twice 
the  size,  and  size  does  not  affect  the  E.  M.  F.  (Art.  633).  The  re- 
sistance, however,  is  only  half  that  of  a  single  cell,  because  the 
cross-section  of  the  liquid,  which  the  current  has  to  pass,  is  twice 
as  great.  For  two  cells  in  multiple  arc,  then,  the  current 


We  can  extend  this  reasoning  to  any  number  of  cells,  and  say, 
n  cells,  in  series,  multiply  the  E.  M.  F.  and  the  internal  resistance 

by  n,  and  m  cells,  in  multiple  arc,  divide  the  internal  resistance  by 

m,  but  leave  the  E.  M.  F.  unaltered,  it  being  that  of  a  single  cell. 

In  general,  if  we  have  m  n  cells,  consisting  of  n  groups  in  series, 

each  group  containing  m  cells  in  multiple  arc,  the  resulting  cur- 

rent will  be 

c=        nE 

E  +  n  — 
m 


408  ELECTRICITY     AND     MAGNETISM. 

With  a  given  number  of  cells  and  a  given  external  resistance- 
some  arrangement  of  the  cells  can  be  found  which  will  give  a  maxi- 
mum current.  It  can  be  proved  that  this  arrangement  will  ren- 
der the  internal  resistance  as  nearly  equal  to  the  external  as  possible. 

When  the  external  resistance  is  very  great  compared  with  the 
battery,  it  is  advisable  to  get  as  much  E.  M.  F.  as  possible.  This 
is  accomplished  by  placing  the  cells  in  series. 

A  Problems. 

^  V 

1.  An  incandescent  lamp  takes  a  current  of  0.7  ampere,  and  the 
E.  M.  F.  between  its  terminals  is  found  to  be  98  volts  :  what  is  its- 
resistance  ? 

2.  A  current  of  8.5  amperes  flows  through  a  conductor,  the 
ends  of  which  are  found  to  have  a  difference  of  potential  of  24 
volts  :  what  is  its  resistance?    yj.  *  R  Ans.  2.823  ohms. 

3.  A  battery,  arranged  in  series,  consists  of  5  Daniell  cells,  each 
having  an  E.  M.  F.  of  1.08  volt  and  an  internal  resistance  of  4 
ohms  :  what  current  will  the  battery  produce  with  an  external  re- 
sistance of  7  ohms.  -  ---4ns.  0.2  ampere. 

J  4.  Two  cells  of  E.  M.  Ft,  1.8  volt  and  1.08  volt  respectively, 
are  placed  in  circuit  in  opposition  (i.e.,  with  their  poles  in  such 
positions  that  the  cells  tend  to  send  currents  in  opposite  direc- 
tions). The  current  is  found  to  be  0.4  ampere  :  what  current 
will  be  produced,  if  the  cells  are  placed  properly  in  series? 

Ans.  1.6  ampere. 

5.  A  Bunsen  cell  has  an  internal  resistance  of  0.3  ohm  and 
its  E.  M.  F.  on  open  circuit  is  1.8  volt.  The  circuit  is  completed 
by  an  external  resistance  of  1.2  ohm  :  find  the  current  produced 
and  the  difference  of  potential  which  now  exists  between  the  ter- 

minals of  the  cell.  A     }  (  C  =  1.2  ampere. 

-Ans.  ^p  D  =144volt 

*  6.  Two  wires  of  the  same  length  and  material  are  found  to  have 
resistances  of  4  and  9  ohms  respectively  :  if  the  diameter  of  the 
first  is  1  mm.,  what  is  the  diameter  of  the  second  ?  "  ~^  = 

7.  The  resistance  of  a  bobbin  of  wire  is  measured  and  found 
(->  fr.     to  be  68  ohms'  :  a  portion  of  the  wire  2  metres  in  length  is  now  cut 
~n~<~     off,  and  its  resistance  is  found   to  be  0.75  ohm.     What  was  the 
total  length  of  wire  on  the  bobbin?'  Ans.  181.3  metres. 

^  8.  What  length  of  platinum  wire  1  mm.  iu  diameter  is  required 
in  order  to  make  a  10  ohm  resistance  coil?  b^ 

Vt  y.  A  wire  ra  metres  in  length  and  I/nth  of  a  millimetre  in  di- 
ameter is  found  to  have  a  resistance  r:  what  is  the  specific  re- 
sistance of  the  material  of  which  it  is  made  ? 


I 


X      A 


THE     CURRENT'S    LINES     OF     FORCE.  409 

10.  A  uniform  wire  is  bent  into  the  form  of  a  square  :  find  the 
resistance  between  two  opposite  corners  in  terms  of  the  resistance 
of  one  of  the  sides.  $  &j 

11.  Twelve  incandescent  lamps  are  arranged  in  parallel  between 
two  electric  light  leads.     The  difference  of  potential  between  the 
leads  is  99  volts,  and  each  lamp  takes  a  current  of  0.75  ampere  : 
what  is  the  equivalent  resistance  between  the  leads  ? 

Ans.  11  ohms. 

12.  A  battery  of  20  ohms  resistance  is  joined  up  in  circuit  witlf  f- 

a  galvanometer  of  10  ohms  resistance.     The  galvanometer  is  then  .  fi  ~{ y 
shunted  by  a  wire  of  the  same  resistance  as  its  own  :  compare  the 
currents  produced  by  the  battery  in  the  two  cases. 

Ans.  C  :  C'  =  5  :  6. 

13.  In  the  preceding  example  determine  the  ratio  between  the 
currents  which  flow  through  the  galvanometer  before  and  after  it 
is  shunted. 

14.  How  would  you  arrange  a  battery  of  12   cells,  each  of  0.6 
ohm  internal   resistance,   so    as  to    send    the    strongest    current 
through  an  electro-magnet  of  resistance  of  0.7  ohm. 

'J  15.  In  a  Wheatstone's  bridge  (Fig.  356)  A  =  10  ohms,  B  — 
1,000  ohms,  and  G  —  50  ohms  :  what  is  the  resistance  of  D,  if  the 
galvanometer  shows  no  current  ?  A  :  8  ;  :  £  •'  £  Ans.  5,000  ohms. 


CHAPTER    VI. 

ELECTRO -MAGNETISM. 

648.  The  Current's  Lines  of  Magnetic  Force.  — If  a 
wire,  carrying  a  current  of  electricity,  be  passed  through  a  sheet 'of 
paper,  as  indicated  in  Fig.  359,  and  if  iron  filings  be  sprinkled  upon 
the  paper,  they  will  arrange  themselves  so  as  to  form  circles  around 
the  wire.  If  then  a  short  magnetic  needle  be  moved  about  the 
wire,  it  will  tend  to  place  itself  tangentially  to  the  circle  passing 
through  its  centre.  If  the  direction  of  the  current  be  reversed, 
the  needle  will  turn  through  180°.  The  circles  of  the  filings  show 
the  paths  of  magnetic  lines  of  force,  which  owe  their  existence  to 
the  electrical  current,  just  as  the  filings  in  Fig.  335  showed  the 
paths  of  the  lines  of  force  of  a  magnet.  In  order  to  give  a  direc- 
tion to  these  circular  lines  we  must  consider  in  what  direction  an 
isolated  north  magnetic  pole  would  move.  In  the  diagram  this 


410 


ELECTRICITY     AND     MAGNETISM. 


would  evidently  be  contrary  to  tbe  motion  of  the  bands  of  a  clock. 
In  general,  to  remember  the  directions  which  these  lines  will  have, 


FIG.  359. 


BATTERY 


#m 


Maxwell  makes  use  of  the  thrust  and  turn  of  an  ordinary  screw 
(Fig.  360).     Suppose  the  current  to  flow  along  the  axis  of  the 

screw,   from   the   head 

FIG.  360.  to  the  point  when  it  is 

being  screwed  into  any- 
thing, and  vice  versa 
when  it  is  being  re- 
moved— i.e.,  the  direc- 
tion of  the  current  is  the  same  as  the  direction  of  propagation  of 
the  screw — then  the  direction  of  the  circular  lines  of  force  is  the 
same  as  the  motion  of  the  circumference  of  the  head  of  the  screw 
when  it  is  screwed  in  or  out. 


649.  Effect  of  a  Current  on  a  Magnet. —  The  experiment 
in  the  preceding  article  shows  that  there  is  a  connection  between 
electricity  and  magnetism.  In  1819,  Oerstedt  showed  that  a 
magnet  tends  to  set  itself  at  right  angles  to  a  wire  carrying 
an  electric  current.  He  found,  further,  that  the  way  in  which 
the  north  end  of  the  needle  turns,  whether  to  the  right  or  left  of 
its  normal  position,  depends  upon  the  position  of  the  wire  that 
carries  the  current — whether  it  is  above  or  below  the  needle — 
and  upon  the  direction  in  which  the  current  flows  through  the 
wire.  The  position  which  a  magnet  will  tend  to  take,  when  under 


SOLENOIDS. 


411 


FIG.  361. 


the  influence  of  a  current,  can  be  easily  determined  by  knowing 
the  direction  of  the  lines  of  force  of  both  current  and  magnet, 
and  by  considering  that  the  magnet  will  move  in  such  a  direc- 
tion as  to  tend  to  bring  its  lines  of  force  into  the  same  path  and 
dii'ection  as  the  lines  of  force  of  the  cur- 
rent. Sometimes  the  student  forgets  the 
direction  of  the  current's  lines.  In  such 
a  case  let  him  remember  that  if  the  cur- 
rent flows  from  South  to  North  and  Over 
the  needle,  the  north  end  of  the  needle 
will  be  turned  toward  the  West,  the  com- 
bination being  remembered  by  the  initial 
letters,  SNOW. 

If  we  suppose  the  magnet  to  be 
fixed,  and  the  conductor  carrying  the 
current  to  be  movable,  then  the  con- 
ductor will  move  because  of  the  strife 
toward  parallelism  of  their  lines  of  force. 
Lodge  illustrates  this  by  a  beautiful  ex- 
periment. Send  a  strong  current  through 
a  vertically  suspended  gold  thread  (such 
as  is  used  upon  military  garments). 
Alongside  the  thread  place,  vertically,  an 
electro-magnet  (Fig.  361).  Upon  excit- 
ing the  magnet  the  thread  will  wind  itself 
around  the  magnet.  Kever.se  the  cur- 
rent and  it  will  unwind  and  then  rewind 
itself  in  the  opposite  direction.  Com- 
plete parallelism  of  the  lines  of  force  is,  of  course,  impossible, 


FIG.  362. 


but  the  experiment  well  illustrates 
the  tendency. 

The  movement  of  a  needle  under 
the  influence  of  a  current  furnishes 
a  convenient  means  of  determining 
the  direction  in  which  the  current 
is  flowing. 

650.  Solenoids.  —  If  a  wire 
which  carries  a  current  be  bent  in- 
to a  circle,  all  the  lines  of  force  will 
emerge  from  one  side  of  an  imag- 
inary disc  bounded  by  the  loop 
(Fig.  362)  and  bending  around  the 
wire  will  enter  the  opposite  side. 
The  loop,  because  of  the  current,  will  be  magnetically  equivalent 


412 


ELECTRICITY     AND    MAGNETISM. 


FIG.  363. 


to  a  disc  magnet  having  north  polarity  on  one  side  and  south 
polarity  on  the  other.     In   the   diagram  an  isolated  north  pole 

placed  above  the  surface  of  the 
page  would  be  attracted  to- 
ward p,  as  though  it  were  a 
south  magnetic  pole.  If  the 
wire  is  coiled  into  the  shape 
indicated  in  Fig.  363,  it  is 
termed  a  solenoid  or  helix. 
Upon  passing  a  current,  the 
lines  of  force,  from  their  mut- 
ual action,  take  the  paths  indicated  in  the  figure.  A  solenoid, 
when  traversed  by  a  current,  has 
the  same  magnetic  effect  as  a  bar 
magnet  whose  axis  coincides  with 
the  axis  of  the  solenoid.  Sole- 
noids exhibit  all  the  properties 
of  magnets — attract  pieces  of  soft 
iron,  attract  and  repel  magnets 
or  other  solenoids,  and,  if  sus- 
pended by  non-restraining  quick- 
silver contacts,  as  in  Fig.  364,  will  turn  into  the  earth's  magnetic 
meridian. 


FIG.  364. 


FIG.  365. 


651.  Ampere's  Theory  of  Magnetism. — Because  of  the 

like  actions  exerted  by  sole- 
noids and  magnets,  Ampere 
concluded  that  the^  perma- 
nent magnetism  of  steel 
owed  itself  to  circular  mole- 
cular currents  of  electricity, 
as  shown  in  Fig.  3(>5.  He 
showed  that  the  resultant 
of  these  many  molecular 
currents  was  equivalent  to 
surface  solenoid al  currents, 
as  indicated  in  Fig.  366.  In  the  interior  of  the  magnet  cur- 
rents on  contiguous  molecules  are  running  in  opposite  directions, 
and  accordingly  neu- 
tralize each  other's 
magnetic  effects.  Half 
of  the  currents  on  the 
surface  molecules  are 

not  neutralized,  and  the  combined  effect  is  the  same  as  a  surface 
solenoidal  current. 


FIG. 


MAGNETO-MOTIVE     FORCE. 


413 


FIG.  367. 


It  should  be  remembered  that  looking  at  the  north  end  of 
a  magnet,  end  on,  the  amperian  currents  run  counter-clockwise. 

652.  Electro-magnets.— We  have  seen  (Art.  611)  that  when 
a  piece  of  iron  is  placed  in  a  magnetic  field,  it  becomes  an  induced 
magnet  and  it  adds  lines  of  force  to  the  field.     Now,  if  an  iron 
core  be  inserted  into  a  solenoid,  it  becomes  a  magnet  under  the 
influence  of  the  solenoid's  field,  and,  because  of  its  much  greater 
permeability  than  air  (Art.  612),  adds  many  lines  of  force  to  the 
field.     Such  a  combination  is  termed  an  electro-magnet.     An  electro- 
magnet differs  from  an  ordinary  one  in  that  the  instant  the  excit- 
ing current  is  removed  the  electro-magnet  loses  its  magnetism. 

The  intensity  of  the  field  which  a  given  solenoid  can  produce  is- 
limited  only  by  the  strength  of  the  current  traversing  it.  If  the 
current  strength  is  doubled,  the  strength  of  the  field  is  doubled. 
The  iron  core,  upon  being  inserted,  multiplies  the  strength  of  the 
field  by  a  certain  factor  (the  permeability  of  the  core,  see  Art. 
612).  Now,  if  the  permeability  of  iron  were  constant,  there  would 
be  scarcely  any  limit  to  the  strength 
which  could  be  given  to  an  electro- 
magnet. But  as  the  iron  reaches  the 
point  of  saturation  its  permeability 
decreases  toward  unity.  As  it  is, 
electro-magnets  can  be  made  many 
times  more  powerful  than  permanent 
magnets. 

A  common  form  of  electro-mag- 
net is  schematically  shown  in  Fig. 
367.  The  solenoid  with  its  core  is 
bent  into  the  form  of  a  horseshoe. 
An  actual  magnet  would  be  wound  t 
with  many  more  turns  of  wire,  which  must,  of  course,  be  insulated 
from  the  core.  Upon  passing  a  current,  a  heavy  weight  can  be 
suspended,  and  this  will  detach  itself  as  soon  as  the  current  is  dis- 
continued. 

653.  Magneto-motive  Force. — In  the  practical  construction 
of   electro-magnetic  apparatus  it   is  often   desirable  to  obtain   a 
maximum  number  of  lines  of  force  in  a  given  region.     This  region 
is  to  be  occupied  by  some  movable  armature  or  object,  to  be  sub- 
jected to  the  field's  influences.     Let  us  consider  how  this  can  be 
obtained  when  this  region  is  the  space  between  the  poles  of  an 
electro-magnet.     Evidently  by  increasing  the  number  of  lines  gen- 
erated by  the  current  and  adding  as  many  lines  as  possible  to  these 
by  proper  selection  of  materials  and  shape  of  the  electro-magnet. 


414  ELECTRICITY     AND     MAGNETISM. 

The  number  of  lines  originally  pi'oduced  will  increase  with  the 
current  strength  which  flows  through  the  solenoid  or  coil.  It 
will  also  increase  with  the  number  of  the  loops  of  the  wire  in  the 
coil — for  each  loop  will  add  the  same  number  of  lines  to  those 
already  traversing  the  axis  of  the  coil.  Accordingly  we  must 
employ  as  strong  a  current  and  as  many  turns  of  wire  as  pos- 
sible. Remembering  that  the  magnetic  permeability  of  a  sub- 
stance is  the  same  as  its  conductivity  toward  lines  of  force,  it  is 
desirable  that  all  the  space  which  is  traversed  by  lines  of  force, 
except  the  portion  which  is  to  be  employed  for  the  movement 
of  the  object  to  be  subjected  to  the  field's  influence,  should  be 
occupied  by  a  substance  of  maximum  magnetic  permeability,  i.e., 
by  the  best  soft  iron. 

Now  we  can  obtain  a  law  for  the  flow  of  lines  of  magnetic  force 
exactly  like  Ohm's  law  (Art.  641)  for  current  flow.  Call  the  source 
of  the  lines  the  magneto-motive  force  (M.  M.  F.) ;  call  the  reciprocal 
of  the  conductivity  the  magnetic  resistance  (E) ;  then  the  magnetic 
flux  or  number  of  lines  which  pass  through  the  axis  of  the  coil 

M.M.F. 
N=^E~ 

Evidently  the  M.  M.  F.  is  &  function  of  the  current  strength  (c) 
and  the  number  of  loops  (n)  made  by  the  coil  wire.  It  can  be  shown 
mathematically  that  if  c  is  expressed  in  amperes,  and  N  is  to  be 

obtained  in  absolute  units,  the  M.  M.  F.  =  -—  n  c.     The  magnetic 

resistance  is  subject  to  the  same  law  as  electrical  resistance  (Art. 
639).  Increase  the  length  I  of  the  path  to  be  travelled  by  the 
lines  and  the  resistance  is  increased.  It  is  decreased  by  increas- 
ing the  cross-section  q,  and  decreases  with  increase  of  magnetic 
permeability  p.,  so  that  the  resistance 

B*>L 

/*  9 
Introducing  these  values  in  the  equation  for  the  flux,  we  obtain 

y -£*£?. 

10  — 

p-q 

This  formula  is  of  great  importance  and  has  been  of  great  service 
in  the  designing  of  efficient  electro-magnetic  machinery,  e.g.,  dy- 
namos and  motors.  To  understand  the  application  of  it,  let  us 
refer  to  Fig.  367.  The  region  where  maximum  flux  is  desired  is 
the  bottom  of  the  horseshoe  where  the  armature  and  the  weight 
attached  to  it  are  suspended.  The  flux  will  be  increased  by 
increasing  the  current  (c),  the  number  of  turns  of  wire  (n),  the  cross- 
sections  (q)  of  the  core,  the  armature  and  the  air  gaps  between  the 


THE     TELEGRAPH.  415 

armature  and  the  poles,  by  increasing  the  permeability  (/A)  of  the 
core  and  armature,  and  by  decreasing  the  average  lengths  (Z)  of  the 
core,  the  armature,  and  the  air  gaps. 

When  it  is  considered  that  the  permeability  of  iron  is  much 
greater  than  that  of  air,  it  will  be  seen  that  the  force  of  attraction 
would  be  greatly  lessened  if  a  piece  of  the  iron  core  were  removed 
from  the  top.  The  force  exerted  by  a  hoi'seshoe  electro-magnet 
is  much  greater  than  that  exerted  by  two  parallel  straight  electro- 
magnets corresponding  to  the  two  legs  of  the  horseshoe. 

654.  The  Morse  Telegraph  System. — The  fact  that  an 
electro-magnet  loses  its  magnetism  as  soon  as  the  exciting  current 
is  discontinued,  was  made  use  of  by  Professor  Morse  in  the  construc- 
tion of  his  system  of  electric  telegraph.  In  this  system  an  oper- 
ator at  one  station  can,  by  making  and  breaking  a  current  of  elec- 
tricity which  traverses  a  wire  to  a  second  station,  produce  or  de- 
stroy, at  will,  the  magnetism  of  an  electro-magnet  in  the  second 
station.  This  electro-magnet  is  a  part  of  an  instrument  called  a 
register,  which  will  be  described  later.  According  as  the  magnet 
is  excited  for  a  longer  or  shorter  interval,  the  register  marks  upon 
a  moving  band  of  paper  a  series  of  dashes  or  dots.  These  may  be 
combined  so  as  to  serve  as  an  alphabet. 

The  Morse  circuit  has  four  elements :  A  battery  to  produce  a 
current  ;  a  key  to  manipulate  the  current ;  a  register  or  sounder  to 
record  the  current  thus  manipulated  ;  and  a  line  to  convey  the 
current. 

The  battery  generally  employed  is  a  modification  of  the  Dan- 
iell's  type,  called  the  Gravity  Battery  (Art.  635).  Dynamos  are, 
however,  rapidly  supplanting  them  in  the  large  telegraph  systems. 

The  key  for  manipulating  the  current  consists  of  a  lever,  B 
(Fig.  368),  and  anvil,  C,  both  of  brass,  and  insulated  from  each 


FIG. 


other.  The  anvil  is  connected  to  one  terminal  of  the  line,  say 
from  the  battery,  and  B  to  the  other  terminal,  where  it  leaves  for 
the  receiving  station.  The  end  of  B  is  depressed  by  the  finger  of 


416 


ELECTRICITY     AND     MAGNETISM. 


the  operator  on  the  insulating  button  F,  and  is  raised  by  the  spring 
E,  when  the  pressure  is  removed.  The  former  movement  closes 
the  circuit,  the  latter  opens  it,  and  by  a  succession  of  these  the 
message  is  sent.  When  the  key  is  not  in  use,  the  brass  bar  K, 
hinged  to  the  base  of  B,  is  pressed  into  contact  with  C.  This 
closes  the  circuit  so  that  other  operators  on  the  line  may  have  a 
continuous  circuit  when  they  desire  to  send  a  message.  When  not 
in  use,  the  line  is  traversed  by  a  current. 

The  register  for  recording  the  message  on  paper  is  constructed 
as  follows  : 

The  lever  E  is  furnished  with  a  style  e  (Fig.  369),  directly  over 
which  is  a  groove  on  the  surface  of  a  solid  brass  roller  c.  Between 
c  and  e  is  a  long  paper  ribbon  R  R.  Attached  to  ^  is  a  soft  iron 
armature  A,  placed  above  the  magnet  M,  and  furnished  with  a 


FIG 


spring  s  to  raise  it  as  far  as  the  screw  i  allows  when  it  is  not  at- 
tracted by  M.  When  the  circuit  is  closed,  A  is  attracted  and  e 
rises  and  forces  the  paper  into  the  groove,  producing  a  slight  eler 
vation  on  its  upper  surface.  The  ribbon  is  pulled  along  at  a  uni- 
form rate  in  the  direction  of  the  arrow  by  clockwork  (not  shown 
in  the  figure),  so  that  when  the  circuit  remains  closed  for  a  little 
time,  a  dash  is  marked  on  the  paper  by  e  ;  when  it  is  closed  and 
instantly  opened,  the  result  is  a  dot — or  rather  a  very  short  dash. 
Spaces  are  left  between  these  whenever  the  circuit  is  opened. 
Combinations  of  these  dots,  dashes,  and  spaces,  all  carefully  regu- 
lated in  length,  compose  the  letters  of  the  alphabet.  Spaces  are 
also  left  between  the  letters,  and  longer  ones  between  words. 

By  lengthening  the  circuit  wire,  it  is  evident  that  the  person 
who  sends  the  message  at  n  p,  and  the  one  who  receives  it  at  E, 
may  be  miles  apart,  and  the  transmission  will  be  almost  instanta- 
neous, owing  to  the  rapid  passage  of  the  current. 

It  has  been  found  that  the  ear  is  sufficiently  accurate  to  allow 
of  the  dispensing  with  the  register,  as  used  by  Morse.  Instead  of 


THE     RELAY. 


417 


it  a  sounder  is  employed.     In  this  the  end  of  the  lever  L'  (Fig.  370), 
instead  of  being  furnished  with  a  style,  is  made  to  strike  against 


the  two  screws,  N',  O'.  The  downward  click  is  a  little  louder  than 
the  upward  one,  and  so  the  beginning  and  end  of  each  dot  or 
dash  are  distinguished  from  each  other.  Many  operators  learn 
from  the  first  to  read  by  the  ear, 'and  have  never  used  a  register. 

For  a  line  it  was  at  first  supposed  that  a  complete  metallic  cir- 
cuit was  necessary,  hence  a  return  wire  was  employed.  But  this 
was  rejected  when  it  was  found  that  the  earth  could  be  used  as  a 
part  of  the  circuit,  as  shown  in  Fig.  371,  in  which  the  dotted  line 
and  arrow  beneath  the  surface  are  not  intended  to  convey  the  idea 
that  a  current  actually  flows  from  one  earth-plate  to  the  other,  but 

FIG.  371. 


that  a  complete  circuit  is  formed,  the  earth  acting  the  part  of  an 
infinite  reservoir  of  electricity.  S  and  S '  are  the  terminal  stations, 
and  s  is  one  of  the  way  stations  which  may  occur  anywhere  along 
the  line.  At  every  station  both  a  key,  G,  and  sounder,  7,  are  in- 
troduced into  the  circuit,  so  that  messages  can  be  both  sent  and 
received. 

655.  The   Relay. — When  a  telegraph  line  is  very  long,  its 
resistance  is  high  and  the  leakage,  because  of  insufficient  insula- 


418  ELECTRICITY     AND     MAGNETISM. 

tion,  is  great.  Hence  a  current  sufficiently  strong  to  satisfactorily 
operate  a  register  or  sounder  cannot  be  economically  sent  through 
it.  Accordingly  use  is  made  of  a  relay.  In  this  instrument  (Fig. 
372)  the  line  current  entering  at  W  and  leaving  at  W"  excites  the 
electro- magnet  M.  This  attracts  the  armature  A  of  the  delicately 
adjusted  lever  L.  The  adjustment  is  obtained  by  regulating  the 
tension  exerted  by  the  spiral  spring  s.  During  the  passage  of  a 
current  along  the  line,  the  lever  L  plays  lightly  to  and  fro,  but 
with  insufficient  strength  to  act  as  a  register  or  sounder.  It  can, 
however,  be  made  to  act  as  a  key  for  a  separate  local  circuit  in  the 
receiving  office.  One  terminal  of  this  local  circuit,  which  contains 
a  sounder  and  battery,  is  connected  by  the  binding-post  /  with  the 

\ 

FIG.  372. 


lever  L.  The  other,  through  n,  is  connected  with  the  screw  N. 
When  the  distant  operator  closes  his  key  the  armature  A  causes 
the  lever  L  to  close  the  local  circuit  at  N.  When  the  distant 
operator  opens  his  key,  the  spring  s  opens  the  local  circuit,  Thus 
the  moving  lever  of  a  relay  acts  as  a  key  for  a  local  circuit. 

Evidently  the  relay  may  be  used  for  repeating  a  message  on 
another  long  circuit. 

656.  Duplex  Telegraphy. — In  the  Morse  system  just  de- 
scribed evidently  but  one  message  can  traverse  the  wire  at  the 
same  time.  If  two  could  simultaneously  traverse  it,  the  earning 
capacity  of  the  line  would  be  doubled.  This  feat  can  be  accom- 
plished, and  is  termed  duplex  telegraphy.  A  simple  duplex  system, 
employing  the  principle  of  Wheatstone's  bridge  (Art.  64G),  is 
shown  in  Fig.  373,  which  represents  two  stations  connected  by 
the  line  wire  L  L'.  C  L  11  is  a  Wheatstone  bridge,  modified  to 
suit  the  conditions  of  the  case,  /  the  sounder,  R  resistance  coils,  k 
a  key  working  upon  the  centre  and  having  forward  and  back  con- 


DUPLEX     TELEGRAPHY. 


tacts  at  a  and  c,  b  the  battery,  and  E  the  earth  connections.     The 
same  letters,  accented,  represent  like  parts  at  the  second  station. 


FIG.  373. 


C*' 


When  not  in  use  the  keys  make  back  contact  by  the  action  of  a 
spring.  The  ratio  of  the  resistance  C  R  and  E  E3  is  made  equal  to 
that  of  C  L  and  the  line  wire  L  L',  including  the  back  contact  earth 
connection  at  the  second  station.  When  thus  balanced  any  current 
arriving  at  C,  which,  dividing,  passes  through  C  L  L'  and  C  R  E3, 
will  maintain  the  points  L  and  R  at  the  same  potential. 

If  now,  a'  being  closed,  a  be  closed,  a  current  will  now  through 
a  and  k  to  C,  where  it  will  divide,  one  part  going  to  earth  through 
R  and  E3,  and  the  other  through  L  L'.  As  the  potentials  were 
made  equal  at  L  and  It,  no  current  will  pass  through  the  indicator 
/;  that  part  of  the  current  which  flows  through  L  L'  divides  at  L', 
part  going  through  C'  k'  a  to  E",  and  part  through  /'  (giving- 
signal)  and  R'  to  E'.  Thus  the  closing  of  a  gives  a  signal  at  1' 
but  none  at  /. 

If  now  the  second  operator  should  close  his  key  while  a  was 
closed,  a  current  from  b'  would  flow  through  c  and  k'  to  C',  where 
it  would  divide,  part  going  to  earth  through  R'  and  E'  (joining 
the  current  already  flowing  through  from  L  L'),  and  part  would 
flow  to  L'  and  oppose  the  current  from  the  other  station  ;  this 
opposing  current  will  have  the  same  effect  as  increased  resistance 
in  the  line  wire  L  L',  and  hence  the  balance  C  L  R  will  be  dis- 
turbed, the  potential  of  L  rising  above  that  of  R,  and  resulting  in 
a  current  from  L  through  /  to  7?,  giving  a  signal  at  /.  Thus  the 
register  at  each  station  will  respond  to  the  key  of  the  other,  and 
only  to  that,  whether  one  or  both  operators  be  signalling. 

The  above  explanation  of  the  principle  of  this  particular  mode 
of  sending  simultaneous  messages  in  opposite  directions  on  a 
single  wire,  does  not  pretend  to  describe  the  actual  arrangement 
of  wires  or  earths  in  use.  For  a  full  description  of  the  various 
modes  of  duplex  and  quadruplex  telegraphy  the  student  is  referred 
to  works  on  practical  telegraphy. 


420  ELECTRICITY     AND     MAGNETISM. 

657.  Atlantic  Telegraph  Cable.  — This  cable  stretches  a 
distance  of  3,500  miles,  and  from  the  nature  of  the  case  is  a  con- 
tinuous wire,  so  that  it  cannot  be  advantageously  worked  by  the 
Morse  apparatus.     The  indicator  employed  is  a  sensitive  galva- 
nometer needle,  which  is  made  to  oscillate  on  opposite  sides  of  the 
zero  point  by  the  passage  through  it  of  currents  in  opposite  direc- 
tions.    But  to  reverse  the  direction  of  the  current  throughout  the 
whole  length  of  the  cable  is  a  slow  process.     For  the  cable  is  an 
immense  Leyden  jar,  the  surface  of  the  copper  wire  (amounting  to 
425,000  sq.  feet)  answering  to  the  inner  coating,  the  water  of  the 
ocean  to  the  outer,  and  the  gutta-percha  between  the  two  to  the 
glass  of  an  ordinary  jar.     A  current  passing  into  it  is  therefore  de- 
tained by  electricity  of  the  contrary  kind  induced  in  the  water,  and 
no  effect  will  be  produced  at  the  farther  end  until  it  is  charged. 

This  very  circumstance,  at  first  considered  a  misfortune,  is 
now  taken  advantage  of  in  a  very  simple  and  ingenious  manner  to 
facilitate  the  transmission  of  signals.  The  current  is  allowed  to 
pass  into  the  cable  till  it  is  charged — then,  without  breaking  the 
circuit,  by  depressing  a  key  for  an  instant,  a  connection  is  made 
between  it  and  a  wire  running  out  into  the  sea ;  that  is,  between 
the  inner  and  outer  coatings.  This  partially  discharges  it,  and 
the  needle  at  the  other  end  is  deflected.  When  the  key  is  raised 
the  discharge  ceases,  the  current  flows  on  as  before,  and  the 
needle  is  deflected  in  the  opposite  direction. 

658.  Electric  Bells. — The  ordinary  electric  house  bell  con- 
sists of  an  electro-magnet,  which  moves  a  hammer  backward  and 

FIG.  874. 


forward  by  alternately  attracting  and  releasing  it,  so  that  it  beats 
against  a  bell.     The  arrangements  of  the  instrument  are  shown  in 


GALVANOMETERS.  421 

Fig.  374.  A  current  from  a  battery  (usually  of  the  Leclanche 
pattern),  after  traversing  the  electro-magnet  E,  enters  a  spring 
attached  to  the  armature  and  bell-hammer.  It  leaves  the  spring 
by  an  adjustable  screw,  G,  and  returns  to  the  battery.  When  it 
flows  it  excites  the  magnet  which  attracts  the  armature  and  causes 
the  hammer  to  hit  the  bell.  In  moving  toward  the  magnet  the 
contact  at  G  has  been  broken,  and  the  magnet  losing  its  magnetism 
allows  the  armature  to  spring  back  so  that  the  contact  is  renewed. 
This  operation  is  repeated,  the  current  repeatedly  making  and 
breaking  itself.  One  of  the  wires  from  the  battery  to  the  bell  is 
cut  at  the  point  P,  and  a  push  button  is  inserted.  This  is  shown 
in  section  to  the  right.  An  insulating  knob,  P,  when  pressed, 
brings  a  spiral  spring,  which  is  connected  with  one  end  of  the  cut 
wire,  into  contact  with  the  other  end.  The  circuit  being  closed 
thus,  the  bell  commences  to  ring. 

659.  Galvanometers.— These  instruments  are  employed  in 
the  laboratory  for  the  determination  of  nearly  all  electrical  magni- 

FIG.  375. 


tudes.  They  serve  to  detect  the  presence  of  electrical  currents 
and  to  determine  their  strengths  and  directions.  The  principle  of 
their  action  is  electro-magnetic.  Suppose  a  magnetic  needle  (Fig. 
375),  free  to  move  about  a  pivot,  to  lie  in  the  direction  of  the 
earth's  magnetic  meridian.  Suppose  further,  that  it  be  surrounded 
by  a  coil  of  wire,  whose  windings  are  parallel  to  the  axis  of  the 
needle.  If,  now,  an  electrical  current  be  sent  through  the  coil,  it 
will  develop  magnetic  polarity  in  the  coil  so  that,  e.g.,  its  east  side 
will  be  equivalent  to  a  north  pole  and  its  west  side  to  a  south  pole. 
The  needle  will,  under  this  influence,  tend  to  place  itself  in  an  east 
and  west  direction.  It  will  not  quite  attain  this  direction,  for  it  is 
influenced  by  the  earth's  magnetism  at  the  same  time,  and  this 
tends  to  keep  it  in  the  meridian.  Upon  reversing  the  direction  of 
the  current,  the  polarities  of  the  sides  of  the  coil  become  reversed, 
and  the  needle  turns  so  that  its  poles  project  from  opposite  sides 


422  ELECTRICITY     AND     MAGNETISM. 

of  the  coil.  The  side  toward  which  the  north  end  of  the  needle 
turns  determines  the  direction  of  the  current  in  a  given  galvano- 
meter. The  angle  through  which  the  needle  is  deflected  deter- 
mines the  strength  of  the  current  flowing. 

TANGENT  GALVANOMETERS. — If  the  wire  of  a  galvanometer  be 
wound  on  the  circumference  of  a  ring,  whose  diameter  is  at  least 
twelve  times  the  length  of  the  needle  at  its  centre,  the  strengths  of 
currents  causing  different  deflections  will  be  proportional  to  the 
tangents  of  the  corresponding  angles  of  deflection.  Such  an  in- 
strument is  called  a  tangent  galvanometer.  The  reason  for  having  a 
large  diameter  for  the  coil  is  that  those  of  its  lines  of  force,  which 
are  cut  by  the  short  needle  in  its  excursions,  are  then  straight  and 
perpendicular  to  the  earth's  lines.  The  magnet's  pole  is  thus- 
moved  under  the  influence  of  two  forces,  which  act  continuously  at 
right  angles  to  each  other.  The  law  of  the  tangents  then  follows. 

REFLECTING  GALVANOMETERS. —  In  refined  laboratory  measure- 
ments the  determination  of  a  needle's  deflection,  by  observing  the 
movement  of  a  pointer  over  a  divided  scale,  is  inaccurate  and  in- 
convenient. Instead,  a  small  mirror  is  attached  to  the  magnet  and 
the  deflections  are  measured  by  the  different  divisions  of  a  sta- 
tionary divided  scale,  which  are  reflected  from  the  mirror  into  a 
stationary  telescope.  The  arrangement  is  shown  in  Fig.  319. 

A  method,  much  used  in  England,  is  to  have  the  mirror  reflect 
a  ray  of  light  from  a  small  hole  in  an  opaque  chimney  of  a  lamp 
upon  a  stationary  scale.  The  method  is  very  inconvenient,  as  it  re- 
quires the  observations  to  be  made  in  a  darkened  room.  The  ac- 
curacy to  be  obtained  is  not  as  great  as  by  means  of  a  telescope 
and  scale. 

BALLISTIC.  GALVANOMETERS. — In  many  determinations  it  is  re- 
quired to  measure  currents  which  last  but  for  an  instant,  or  to 
measure  quantities  of  electricity.  The  difficulties  connected  with 
these  determinations  are  much  lessened  if  the  time  required  by 
the  galvanometer  needle  to  make  a  single  oscillation  be  very  great, 
as  compared  with  the  time  occupied  by  the  electricity  in  passing. 
Thus  galvanometers  whose  needles  have  periods  of  from  five  to 
twenty-five  seconds  are  used,  and  are  called  ballistic  galvanometers. 

DIFFERENTIAL  GALVANOMETERS. — These  instruments  are  supplied 
with  two  sets  of  coils,  which  are  so  placed  that  they  will  produce 
the  same  electro-magnetic  effect  upon  the  single  needle,  providing 
they  be  traversed  by  currents  of  the  same  strength  and  direction. 
By  means  of  this  instrument  a  current  in  one  coil  may  be  brought 
to  a  given  strength  by  being  made  to  neutralize  the  effect  upon 
the  needle  from  another  current,  which  is  of  constant  (the  required) 
strength  and  passes  through  the  other  coil  in  an  opposite  direction. 


CHAPTER    VII. 


FIG.  376. 


ELECTRO-DYNAMICS. 

660.  Movement  of  Conductors  Carrying  Currents. — In 

the  preceding  chapter  it  has  been  shown  that  a  conductor  carrying 
an  electrical  current,  and  placed  in  the  vicinity  of  a  magnet,  tends 
to  move  the  magnet,  so  that  the  lines  of  force  from  each  may  be- 
come parallel,  or,  if  the  magnet  be  stationary,  the  conductor  strives 
to  move,  to  attain  the  same  end.  As  might  be  expected,  two 
neighboring  conductors,  while  traversed  by  currents,  tend  to  move 
so  as  to  render  their  lines  of  force  parallel. 

"Without  any  knowledge  of  the  existence  or  properties  of  lines 
of  force,  Ampere,  in  1821,  arrived,  by  experiment,  at  the  following 
laws,  which  could  easily  have  been  predicted  by  such  a  knowledge. 

661.  Parallel  Currents.— 

1.  If  galvanic  currents  flow  through  parallel  wires  in  the  same 
direction,,  they  attract  each  other;  if  in  opposite  directions,  they 
repel  each  other.  These  ef- 
fects are  shown  by  suspend- 
ing wires,  bent  as  in  Fig. 
376,  so  that  their  lower  ends 
may  dip  into  four  separate 
mercury  cups,  a,  b,  a',  6',  by 
means  of  which  connection 
between  the  wires  C  and  D 
and  the  battery  may  be 
readily  made.  The  sus- 
pending threads  should  be 

two  or  three  feet  long,  and  the  mercury  cups  should  be  large 
enough  to  allow  considerable  lateral  movement  of  the  wires.  If 
simultaneous  currents  be  sent  through  the  two  wires  C  and  D,  in 
the  same  direction,  the  wires  will  move  toward  each  other;  if 
currents  be  sent  through  the  wires  in  opposite  directions  at  the 
same  time,  they  will  separate  more  widely. 

Hence,  when  a  current  flows  through  a  loose  and  flexible  helix, 
each  turn  of  the  coil  attracts  the  next,  since  the  current  moves  in  the 
same  direction  through  them  all.  In  this  way  a  spiral  suspended 
above  a  cup  of  mercury,  so  as  to  just  dip  into  the  fluid,  will  vibrate 
up  and  down  as  long  as  a  current  is  supplied.  The  weight  of  the 
helix  causes  its  extremity  to  dip  into  the  mercury  below  it ;  this 
closes  the  circuit,  the  current  flows  through  it,  the  spirals  attract 


424 


ELECTRICITY     AND     MAGNETISM. 


each  other,  and  lift  the  end  out  of  the  mercury ;  this  breaks  the 
circuit,  and  it  falls  again,  and  thus  the  movement  is  continued. 

2.  If  currents  flow  through  two  wires  near  each  other,  which 
are  free  to  change  their  directions,  the  wires  tend  to  become  paral- 
lel to  each  other,  with  the  currents  flowing  in  the  same  direction. 
Thus,  two  circular  wires,  free  to  revolve  about  vertical  axes,  when 
currents  flow  through  them,  place  themselves  by  mutual  attrac- 
tions in  parallel  planes,  as  in  Fig.  377,  or  in  the  same  plane,  as  in 

FIG.  378. 


d    £ 


Fig.  378.  In  the  latter  case,  we  must  consider  the  parts  of  the 
two  circuits  which  are  nearest  to  each  other  as  small  portions  of 
the  dotted  straight  lines,  c  d  and  e  f. 

It  appears,  therefore,  that  galvanic  currents,  by  mutual  attractions 
and  repulsions,  tend  to  place  themselves  parallel  to  each  other  in  such 
a  manner  that  the  flow  is  in  the  same  direction. 

The  force  exerted  between  two  parallel  portions  of  circuits  is 
proportional  to  the  product  of  the  current  strengths,  to  the  length 
of  the  portions,  and  inversely  proportional  to  the  distance  between 
them.  The  force  exerted  by  each  current  acts  in  a  direction  per- 
pendicular to  the  direction  of  the  current. 


Currents  not  Parallel. — Currents,  both  of  ivhich  flow 
toward  a  common  point,  or  both  of  which  flow  away  from  a  common 
point,  attract  each  other. 

If  one  of  two  currents  flows  toward,  and  the  other  away  from  a 
common  point,  the  two  currents  repel  each  other. 

These  cases  are  evident  deductions  from  the  preceding  para- 
graph. Suppose  the  two  currents  (Fig.  379)  to  flow  in  A  and  B 
as  though  they  came  from  C,  then  the  tendency  of  the  wires  A  and 
B  is  towards  parallelism,  and  as  we  suppose  the  currents  to  flow 
from  the  direction  C,  the  wires  must  tend  to  move  toward  each 


ELECTRODYNAMIC    ROTATION. 


425 


other  in  order  to  become  parallel.  The  same  effect  would  be  pro- 
duced if  the  currents  in  A  and  B  were  to  flow  towards  G.  But  if 
the  current  in  A  flows  from 

the  direction  C,  and  that  in  B  FIG.  379. 

towards  the  point  C,  then  the 
tendency  of  the  wires  to  be- 
come parallel,  with  the  cur- 
rents flowing  in  the  same 
direction,  causes  B  to  revolve 

about  C  as  a  centre  till  it  reaches  thg  position  B',  and  then  the 
condition  that  the  currents  shall  flow  in  the  same  direction  will  be 
fulfilled.  It  is  not  necessary  that  we  should  regard  A  and  B  as  ly- 
ing in  the  same  plane. 

A  sinuous  current  produces  the  same  effect  as  a  straight  cur- 
rent having  the  same  general  dii'ection  and  length.  If  a  conductor, 
having  one  portion  sinuous  and  the  other  straight,  be  bent  as  in 
Fig.  380,  so  that  the  current  may  flow  from  a  to  6  through  the 

FIG.  380 


FIG.  381. 


straight  part,  and  from  b  to  c  through  the  sinuous  part,  the  two 
portions  of  the  current  thus  flowing  close  together  in  opposite  di- 
rections, the  joint  electro-dynamic  effect  upon  a  movable  conductor 
parallel  to  a  b  will  be  inappreciable. 

663.  Continuous  Rotation  Produced  by  Mutual  Action 
of  Currents. — Suppose  a  continuous  current  to  flow  through  a 

wire  A,  as  indicated  in  Fig. 
381,  and  that  a  wire  7>,  so 
bent  as  to  dip  into  the  mer- 
cury cup  ?»  at  one  end,  and 
into  the  annular  mercury 
trough  n  at  the  other,  be 
suspended  at  the  middle,  a 
counterpoise,  (7,  keeping  it 
balanced. 

If,  now,  a  current  be  made 
to  flow  from  the  cup  m, 
through  B,  and  thence  out 
again  by  means  of  the  mercury  contact  in  ??,  the  wire  B  will  rotate 
in  a  direction  opposite  to  that  of  the  current  in  A  ;  for  the  current 
in  B,  and  that  in  the  part  of  A  to  the  rig] it.  of  n,  are  both  flowing 
towards  n  and  hence  attract,  while  the  current  in  B  and  that  part 


420 


ELECTRICITY     AND     MAGNETISM 


of  the  current  in  A  immediately  to  the  left  of  ??  are  flowing  in 
directions  to  cause  repulsion. 

A  beautiful  experiment,  illustrating  continuous  rotation,  is  to 
place  a  round,  shallow  dish,  containing  mercury,  on  the  pole  of  a 
vertical,  straight  electro-magnet.  Excite  the  magnet  and  dip  the 
terminals  of  a  circuit,  carrying  a  strong  current,  into  the  mer- 
cury at  the  centre  and  side  of  the  dish  respectively.  A  portion  of 
the  mercury  carries  the  current  from  the  centre  to  the  edge  of  the 
dish.  In  doing  so  it  is  made  to  rotate  by  the  action  of  the  lines  of 
force  from  the  magnet.  As  soon  as  it  has  rotated  a  new  portion 
of  the  mercury  is  made  to  carry  the  current.  This,  in  turn,  gives 
•way  to  another  portion,  and  the  whole  body  of  mercury  is  soon  set 
into  rapid  rotation.  Centrifugal  force,  resulting  from  the  rotation, 
causes  the  mercury  to  heap  up  around  the  edges  of  the  dish,  and 
to  be  depressed  at  the  centre. 


FIG.  382. 


664.  Electro-dynamometer. — This  instrument,  invented  by 
Weber,  is  used  for  measuring  the  strengths  of  electrical  currents. 
Its  action  depends  upon  the  electro-dynamic  attractions  discussed 
in  Art.  661.  The  principles  of  its  construction  are  shown  in  the 
crude  apparatus  represented  in  Fig.  382.  This  consists  of  a  fixed 
hollow  coil  of  wire,  in  the  centre  of  which 
is  suspended  another  smaller  coil.  The 
suspension  is  made  by  means  of  two  fine 
parallel  wires,  placed  one  or  two  milli- 
metres from  each  other.  The  upper  ends 
of  these  wires  are  connected  to  two  in- 
sulated binding-posts,  and  the  lower  ends 
are  connected  with  the  terminals  of  the 
suspended  coil.  The  suspension  is  so 
arranged  that,  when  no  current  is  pass- 
ing through  the  dynamometer,  the  planes 
of  the  two  coils  are  perpendicular  to  each 
other.  If,  now,  a  current  of  electricity 
be  sent  through  the  apparatus  (in  the  fol- 
lowing order  through  the  external  coil, 
down  one  suspension  wire,  through  the 
inner  coil  and  up  the  other  suspension 
wire),  the  suspended  coil  will  turn  and 
strive  to  cause  a  parallelism  of  the  planes 

and  currents  of  both  coils.  The  turning  force  of  the  currents  is 
resisted  by  an  increasing  force  exerted  by  the  twisted  wire  sus- 
pension. With  a  certain  current  the  coil  will  be  deflected  a 
certain  amount — i.e.,  until  the  two  opposing  forces  are  equal. 


INDUCED     CURRENTS.  427 

With  a  stronger  current  the  deflection  will  be  greater.  Thus  the 
magnitude  of  the  deflection  can  serve  as  a  measure  of  the  current 
strength. 

A  peculiarity  of  the  electro-dynamometer  is  that  it  serves  to 
measure  alternating  currents,  i.e.,  those  which  change  their  direc- 
tion, perhaps,  several  thousand  times  per  minute,  equally  as  well 
as  continuous  currents.  A  change  in  the  direction  of  flow  of  the 
main  circuit  changes  the  direction  in  both  coils.  This  does  not 
alter  the  direction  of  the  deflection. 


CHAPTER    Till. 

ELECTRO-MAGNETIC    INDUCTION. 

665.  Currents  of  Electricity  Produced  by  Induction.— 
It  has  been  shown  that  when  a  current  of  electricity  flows  through 
a  conductor  the  air  or  other  dielectric  which  surrounds  the  con- 
ductor is  traversed  by  lines  of  force.     The  presence  of  these  lines 
indicates  that  the  dielectric  is  under  some  sort  of  a  strain.     To  pro- 
duced this  strain  energy  must  have  been  expended  by  the  current 
when  it  commenced  to  flow.     During  the  short  time  that  the  strain 
is  being  produced  there  is  an  opposition  to  the  exciting  current, 
which  is  equivalent  to  a  current  in  an  opposite  direction.     Now,  it 
is  reasonable  to  suppose  that,  if  lines  of  force  or  a  magnetic  field 
be  produced   by  some  agency  around  a  closed  circuit  which  is 
primarily  traversed  by  no  current,  a  current  will  be  produced  in 
this  circuit.     The  direction  will  be  opposite  to  that  which  would 
be  necessary  to  create  the  field,  and  will  last  only  for  the  time  nec- 
essary to  produce  the  strain.     Furthermore,  upon  destroying  the 
field  it  is  reasonable  to  suppose  that  the  energy,  which  it  represents 
will  appear  as  a  current  in  the  same  direction  as  one  which  could 
produce  the  field.     These  suppositions  are  substantiated  by  experi- 
ment, as  was  first  shown  by  Faraday.     The  currents  are  called 
induced  currents    (not   to    be    confounded    with   induced   electro- 
static chai'ges),  and  those  currents  whose  directions  are  the  same  as 
a  current  which  could  produce  the  field  are  termed  direct  currents, 
while  those  in  an  opposite  direction  are  called  inverse  currents. 

666.  Methods  of  Producing  the  Inducing  Field.— Inas- 
much as  induced  currents  are  produced  by  any  variation  in   the 
strength  of  the  field  around  the   conductor  which  carries  them, 
they  can  be  produced  either  by  varying  the   strength  of  the  field 


428 


ELECTRICITY     AND     MAGNETISM. 


FIG.  383. 


current  or  by  moving  the  conductor  into  fields  of  various  strengths. 
For  the  sake  of  clearness  suppose  that  we  are  supplied  with  the 

apparatus  represented  in 
Fig.  383.  c  is  the  primary 
coil  of  wire  which  produces 
the  field,  and  is  traversed 
by  a  current  from  the  bat- 
tery. The  secondary  coil,  in 
which  induced  currents  are 
to  be  produced,  is  repre- 
sented at  d.  Its  terminals 
are  connected  with  a  gal- 
vanometer, which  indicates 
the  presence  and  direction 
of  the  induced  currents. 
Now  suppose  that  c  be 
placed  inside  of  d.  Upon 
starting  the  current  in  c  an 
inverse  current  will  be  in- 
duced in  d,  and  upon 

stopping  it  a  direct  current  will  be  induced.  Permitting  the  cur- 
rent in  c  to  flow,  increasing  or  decreasing  its  strength  will  produce 
inverse  or  direct  induced  currents  respectively.  If  the  current 
strength  in  c  be  maintained  constant,  removing  the  coil  c  will  pro- 
duce a  direct  current,  and  replacing  it  an  inverse  induced  current. 
Induced  currents  may  also  be  produced  by  magnets.  Consider 
a  magnet  to  be  the  equivalent  of  a  solenoid  traversed  by  a  current 
(Art.  651).  Dispensing  with  the  battery  we  have  the  apparatus  in- 


FIG.  384 


FIG.  385. 


dicated  in  Fig.  384.     An  inverse  current  will  be  induced  by  the  in- 
troduction of  the  magnet  into  the  secondary  coil,  and  a  direct  cur- 


SELF-INDUCTION.  42£ 

rent  upon  removing  it.  An  inverse  current  may  also  be  induced  by 
strengthening  the  field  of  a  magnet  which  is  stationary  within  the 
secondary,  by  bringing  a  piece  of  iron  near  to  it.  In  this  case  the  iron 
becomes  a  magnet  by  induction,  as  shown  in  Fig.  385,  and  adds  its 
lines  of  force  to  the  field.  Direct  induced  currents  will  follow  the 
removal  of  the  iron. 

It  is  well  to  remark  that,  as  motion  is  merely  relative,  it  is  im- 
material whether  a  magnet  be  placed  in  a  secondary  coil  or  the  lat- 
ter be  placed  around  the  magnet. 

The  facts  which  have  been  mentioned  may  be  summed  up  in  a- 
single  law  : 

Inverse  induced  currents  always  result  from  an  Increase  in  the 
number  of  lines  of  force  which  pass  through  the  circuit,  and  Direct 
induced  currents  always  result  from  a  Decrease  in  the  number  of 
these  lines. 

667.  Lenz's  Law. — If  two  conductors,  A  and  B,  in  one  of  which, 
A,  a  current  is  flowing,  be  made  to  change  their  relative  positions,  then, 
a  current  will  be  induced  in  B  in  a  direction  which  will  cause  a  mu- 
tual action  in  the  two  conductors  tending  to  oppose  their    motion. 
Thus,  if  A  and  B  be  brought  nearer  together  an  inverse  current  will 
flow  in  B,  and  currents  flowing  in  opposite  directions  repel  each 
other ;  and  if  A  and  B  be  caused  to  move  apart,  then  a  direct  sec- 
ondary current  will  flow  in  B,  and  currents  flowing  in  the  same 
directions  attract  each  each  other.     This  statement  of  the  results 
of  experiments  will  aid  the  memory  in  regard  to  the  directions  of 
the  primary  or  secondary  currents. 

668.  Self-induction. — Whenever  a  current  is  started  in  a  coil 
of  wire,  lines  of  force  are  created  which  increase  in  number  from 
zero  to  a  maximum.     Owing  to  the  increase,   they  induce  cur- 
rents in  the  coil  which  are  opposite  to  the  direction  of  the  original 
current.     Upon  stopping  the  original  current  the  lines   of  force 
decrease  in  number  and  thus  induce  a  direct  current  in  the  coil. 
The  induction  in  such  a  case  is  termed  self-induction,  and  the  cur- 
rents are  termed  extra  or  self-induced  currents. 

The  existence  of  self-induced  currents  may  be  demonstrated 
by  the  Wheatstone  bridge  combination  (Art.  646).  Let  three  of 
the  arms  of  the  bridge  be  made  up  of  resistances  without  self-in- 
duction (Art.  645),  the  fourth  arm  consisting  of  an  ordinary  unifilar 
coil.  For  the  purpose  of  increasing  the  self-induction  of  this  fourth 
arm,  insert  a  piece  of  soft  iron  in  the  coil.  Obtain  a  balance  in  the 
bridge  by  employing  a  constant  current.  When  a  balance  has 
been  obtained  the  galvanometer  will  show  no  deflection.  If  the 


430  ELECTRICITY     AND     MAGNETISM. 

current  be  now  stopped,  the  current  induced  in  the  fourth  arm  will 
cause  a  deflection  of  the  galvanometer  needle. 

669.  Coefficients  of  Mutual  and  Self-induction.  —  It  can 
be  proved  mathematically  that  the  electro-motive  force  induced  in  a 
closed  circuit  is  equal  to  the  rate  of  variation  of  the  number  of  lines 
of  force  which  pass  through  it. 

If  in  a  short  interval  of  time  dt,  the  number  of  lines  of  force  JV 
increases  a  small  amount  dN,  then  the  electro-motive  force 


"will  be  induced  in  the  circuit  which  surrounds  these  lines.  In  case 
two  coils,  a  primary  and  secondary,  be  fixed  in  position,  and  the 
strength  of  the  current  in  the  primary  be  increased  by  an  amount  do 
in  the  short  time  dt}  then  the  electro-motive  force 

*=-*! 

will  be  induced  in  the  secondary  during  that  time.  M  is  a  constant 
•which  is  called  the  coefficient  of  mutual  induction  between  the  two 
coils.  Its  value  depends  upon  the  shape  and  number  of  windings 
of  wire  around  the  respective  coils  and  their  relative  positions.  It 
is  numerically  equal  to  the  number  of  absolute  lines  of  force  which 
•would  be  sent  through  either  coil  when  an  absolute  unit  current  of 
electricity  was  sent  through  the  other  coil.  It  makes  no  difference 
which  coil  be  chosen  as  a  primary  in  determining  M. 

If  it  be  supposed  that  the  two  coils  be  made  to  coincide,  i.e., 
that  there  be  but  one  coil,  then  the  electro-motive  force  of  self- 
induction 


The  constant  L  is  called  the  coefficient  of  self-induction,  and  is 
equal  to  the  number  of  absolute  lines  of  force  which  a  coil  would 
send  through  itself  if  it  were  traversed  by  an  absolute  unit  of  current. 

670.  Induced  Currents  from  the  Earth.  —  If  a  coil  (whose 
terminals  are  connected  with  a  sensitive  galvanometer)  be  placed 
so  that  its  axis  is  parallel  with  the  axis  of  a  dipping-needle  (Art. 
621),  it  will  be  pierced  by  a  maximum  number  of  the  earth's  lines 
of  magnetic  force.  If  it  be  now  turned  through  90°  around  an 
axis  perpendicular  to  its  own  axis,  the  number  of  the  lines  piercing 
it  will  decrease  to  zero,  and  the  galvanometer  will  indicate  that  a 
current  is  being  induced  by  the  rotation. 

Continuous  variations  in  the  strength  of  the  earth's  magnetism 
sometimes  induce  currents  of  considerable  strengths  in  long  tele- 
graphic circuits.  Such  currents  are  known  as  earth  currents. 


INDUCTION     COILS.  431 

671.  Arago's  Rotations. — In  1824  Arago  observed  that  the 
oscillations  of  a  magnetic  needle  were  reduced  in  number  by  sus- 
pending a  copper  plate  above  it.     This  observed  phenomenon  soon 
led  him  to  the  discovery  that  if  a  horizontal  copper  disc  be  made 
to  rotate  rapidly,  a  magnetic  needle  suspended  above  it  Avould  ro- 
tate also.     This  effect  may  also  be  produced  with  other  metals 
though  in  less  degree. 

If  a  disc  of  copper  be  set  spinning  on  an  axis,  between  the 
poles  of  a  powerful  electro-magnet  whose  circuit  is  broken,  the  axis 
of  the  disc  being  parallel  to  the  lines  of  force,  the  rotation  con- 
tinues with  slight  loss  of  velocity  for  a  long  time  ;  but  if  the  circuit  be 
suddenly  closed  the  rotation  is  at  once  checked,  or  possibly  stopped. 
If  such  a  disc  be  kept  in  rapid  rotation  by  a  suitable  band  and 
pulley,  after  the  circuit  is  closed,  the  disc  will  be  heated  by  the 
action  of  the  magnet. 

These  effects  were  explained  by  Faraday  as  being  due  to  cur- 
rents induced  in  the  mass  of  metal.     Thus  let  a  needle,  V  S  (Fig. 
386),  be  suspended  above  a  metal  disc  A  B.     The  magnetic  cur- 
rents flow  around  the  needle  as  indicated  in 
the  figure,  the  currents  below  the  needle  from  FlG-  38G- 

right  to  left  as  shown  by  the  dotted  arrow,  and 
those  above  from  left  to  right,  as  shown  by  the 
full  arrow.  Now  suppose  the  disc  to  be  ro- 
tated in  the  direction  from  A  to  B ;  the  por- 
tions of  the  currents  around  2V  S  which  are 
nearest  to  the  disc  will  induce  in  that  part 
of  the  disc  towards  A  currents  whose  direc- 
tions are  such  as  to  resist  the  motion  of  the 

disc,  according  to  Lenz's  law  (Art  667),  that  is  to  say,  currents 
will  flow  in  the  disc  from  left  to  right ;  while  in  that  part  of  the 
disc  towards  B,  which  is  moving  away  from  2V,  the  induced  cur- 
rents are  from  right  to  left,  and  so  resist  the  motion  of  B  away 
from  TV. 

If  the  needle  had  been  moved,  the  disc  remaining  fixed,  the 
same  analysis  of  the  motion  might  be  made,  and  we  should  find 
that  the  disc  would  resist  the  motion  of  the  needle.  A  copper  col- 
lar or  frame  is  sometimes  used  to  coil  the  galvanometer  wire  upon, 
in  order  to  reduce  or  damp  the  oscillations  of  the  needle,  and  bring 
it  more  quickly  to  rest. 

672.  Induction   Coils. — These   instruments   serve  to    trans- 
form currents  of  low  E.  M.  F.  into  alternating  currents  of  high  E 
M.  F.     Their  forms  and  sizes  are  many,  and  only  their  principle 
need  be  mentioned  here.     A  continuous  current  of  low  E.  M.  F.  is 
passed  through  a  primary  coil  made  of  a  few  turns  of  coarse  insu- 


432 


ELECTRICITY    AND     MAGNETISM. 


FIG.  387. 


lated  copper  wire  (Fig.  387).  The  centre  of  the  coil  is  filled  with 
a  core  of  soft  iron  wires.  Before  passing  through  the  coil  the  cur- 
rent traverses  some  sort 
of  a  current -breaker, 
e.g.,  the  one  shown  in 
the  cut  acts  upon  the 
same  principle  as  the 
breaker  of  the  electric 
bell  described  in  Art. 
658.  By  means  of  this 
breaker  the  current  in 
the  primary  is  rapidly 
made  and  broken.  Al- 
ternating currents  are  thus  induced  in  a  secondary  surrounding 
coil,  which  is  wound  with  many  turns  of  very  fine  insulated  wire. 
The  E.  M.  F.  of  these  induced  currents  is  great  because  the  co- 
efficient of  mutual  induction  is  great.  This  is  owing  to  the  large 
number  of  turns  of  wire  in  the  secondary  and  to  the  presence  of 
the  iron  core.  Both  conspire  to  cause  a  large  number  of  lines  of 
force  to  pierce  the  circuit  during  the  short  interval  required  to' 
make  the  circuit  of  the  primary. 

The  function  of  the  induction  coil,  as  here  given,  is  often  re- 
versed, in  which  case  it  becomes  what  is  termed  a  transformer. 
Transformers  are  much  used  in  the  commercial  distribution  of 
rapidly  alternating  currents  for  lighting  and  other  purposes.  Al- 
ternating currents  of  high  E.  M.  F.  and  low  current  strength  are 
received  from  a  main  line  into  the  finer  wire  coil  of  an  induction 
coil.  The  thick  wire  coil  of  the  transformer  is  connected  with  the 
customer's  home  circuit,  and  delivers  to  it  currents  of  great  strength 
tmt  at  low  potential. 

673.  The  Telephone. — This  instrument  for  reproduction  of 
sound  at  a  distance  by  means  of  electric  currents  is  shown  in  sec- 
tion in  Fig.  388,  in  which  a  a'  is  a  disc  or  diaphragm  of  thin  soft 
iron.tthe  circumference  of 

which  is  firmly  clamped  FIG.  388. 

between  the  mouth  guard 
mm'  and  the  case  n  n', 
upon  the  centre  of  which 
the  sound-waves  from  the 
mouth  impinge,  as  at  E, 
and  communicate  to  it  vi- 
brations corresponding  to 

the  simple  or  composite  sounds  uttered.     These  vibrations  of  the 
disc  cause  a  continual  variation  in  the  distance  of  the  disc  from 


BLAKE     TRANSMITTER. 


433 


the  end  of  a  bar  magnet,  b.  Around  the  end  of  the  magnet  near- 
est to  the  diaphragm  a  a'  is  a  coil,  c  c',  of  fine  insulated  copper 
wire,  the  ends  of  which  are  connected  with  binding-posts,  d  d'. 
From  these  posts  are  carried  wires  to  another  precisely  similar 
instrument  at  the  station  with  which  communication  is  to  be  held. 
When  a  word  is  spoken  into  the  instrument  at  E,  the  vibrations 
communicated  to  the  disc  a  a'  cause  variations  in  the  magnetic 
field  of  the  bar  b,  and  these  variations  induce  electric  currents 
which  flow  in  the  coil  c  c',  and  thence  through  the  connecting 
wires  to  the  coil  in  the  instrument  held  to  the  ear  of  the  listener, 
and  these  currents  in  the  last-named  coil  produce  variations  in 
the  strength  of  the  magnet  of  the  receiving  instrument,  causing 
precisely  the  same  vibrations  in  its  diaphragm  as  were  originally 
set  up  in  the  first.  The  vibrations  of  the  diaphragm  are  trans- 
mitted through  the  air  to  the  ear  ;  and  though  no. sound  has  been 
transmitted  from  one  station  to  the  other,  the  words  spoken  into 
one  instrument  are  distinctly  delivered  by  the  other.  The  sound 
vibrations  are  the  cause  of  electric  currents,  and  these  in  turn 
finally  produce  sound  vibrations  again. 

To  such  perfection  of  action  have  these  instruments  been 
brought,  that  not  only  can  the  spoken  words  be  heard,  but  the 
peculiar  characteristics  of  voice  are  so  faithfully  reproduced  that 
by  these  the  speaker  may  be  recognized. 

674.  The  Blake  Transmitter. — The  electro-motive  forces 
generated  by  the  moving  diaphragm  of  the  Bell  telephone  are 
not  sufficiently  large  to  produce 
satisfactory  results  on  long  lines. 
Therefore  an  instrument  termed  a 
transmitter  is  substituted  for  the 
telephone  at  the  sending  end  of 
the  line.  A  common  and  very 
satisfactory  form  of  transmitter 
is  one  designed  by  Francis  Blake. 
It  is  represented  in  Fig.  389,  and 
its  action  depends  upon  the  prin- 
ciple that  the  electrical  resistance 
offered  by  a  carbon  contact  varies 
greatly  with  the  pressure  exerted 
upon  it. 

The  sound  to  be  transmitted 
is  received  in  the  mouth-piece  M, 

'which  causes  it  to  set  the  diaphragm  D  into  corresponding  vibra- 
tions. Touching  the  rear  of  the  diaphragm  is  a  platinum  or  car- 
bon point,  which  is  attached  to  a  piece  of  watch-spring,  and  which 


FIG.  389. 


434  ELECTRICITY     AND     MAGNETISM. 

is  in  connection  with  one  terminal  of  a  battery,  B.  (This  battery 
is  brought  into  circuit  only  as  the  transmitter  is  to  be  used.)  The 
point  forms  a  loose  contact,  (7,  with  a  carbon  button,  which  is  also 
mounted  upon  a  spring.  The  current  from  the  battery  flows 
through  this  contact  to  the  rest  of  the  circuit.  As  the  diaphragm 
vibrates  it  causes  the  point  to  exert  correspondingly  different 
pressures  upon  the  button.  The  resistance  of  the  circuit  is  thus 
varied,  and  this  results  in  variations  in  the  current  which  are  the 
electrical  counterparts  of  the  sound  vibrations. 

A  Bell  receiver,  placed  in  the  same  circuit  with  the  battery  and 
transmitter,  will  yield,  besides  the  transmitted  sound,  a  disagreeable 
"  sizzling "  noise.  To  obviate  this  an  induction  coil,  /,  is  intro- 
duced. The  varying  current  from  the  battery  and  transmitter  is 
passed  through  the  primary  of  a  small  induction  coil,  and  the  line 
wires,  with  their  receivers  included,  are  connected  with  the  second- 
ary coil.  In  this  case  the  currents  on  the  line  flow  in  opposite 
directions  to  what  they  would,  if  connected  directly  with  the  trans- 
mitter circuit.  This,  however,  is  of  no  consequence. 

The  springs  of  the  transmitter,  which  bear  the  cai'bon  button 
and  platinum  point,  are  fastened  to  one  piece  of  brass,  but  are  insu- 
lated from  each  othei*.  The  amount  of  pressure  at  the  contact  is 
regulated  by  a  screw,  whose  end  hits  the  bent  end  of  the  brass 
holder.  The  holder  is  supported  by  pliable  spring  bands  which, 
are  attached  to  the  case  of  the  transmitter.  Although  this  method 
of  adjustment  is  simple  and  appears  crude,  the  delicacy  of  it  is 
marvellous. 

675.  Dynamos. — These  machines  are  for  converting  mechani- 
cal energy  into  electrical  currents.  A  discussion  of  the  principles 
of  their  consti'uction  is  here  out  of  place,  and  the  student  is  re- 
ferred to  some  one  of  the  many  technical  treatises  on  the  subject. 
The  principle  of  their  action  may  be  described. 

The  dynamo  has  two  essential  parts — a  movable*  conductor, 
called  an  armature,  and  a  magnet,  in  whose  field  the  armature 
moves.  The  armature,  by  its  motion,  varies  the  number  of  the 
field  lines  which  pass  through  it,  and  is  therefore  traversed  by  in- 
duced currents. 

Fig.  390  (taken  from  S.  P.  Thompson's  "  Dynamo-Electric  Ma- 
chinery ")  represents  an  ideal  dynamo  in  its  simplest  form.  N  and 
8  are  the  poles  of  a  field  electro-magnet.  The  lines  of  force  pass 
between  the  poles  and  pierce  the  looped  conductor  C,  which  forms 
the  armature.  The  armature,  in  the  position  indicated  by  the  con- 

*  In  some  machines  the  armature  is  stationary  and  the  field  magnets  are 
movable. 


DYNAMOS. 


435 


tinuous  lines,  is  pierced  by  a  maximum  number  of  the  field  lines  ; 
upon  turning  through  90°,  coming  then  into  the  position  indicated 


FIG.  390. 


l>y  the  dotted  lines,  it  is  pierced  by  none  of  the  lines.  During 
the  whole  quarter  revolution  there  has  been  a  decrease  in  the  num- 
ber of  lines  passing  through  the  loop.  An  induced  current,  flow- 
ing in  a  certain  direction,  has  accompanied  the  movement.  During 
another  quarter  revolution  the  number  of  penetrating  lines  will  be 
on  the  increase,  but  they  now  pass  through  the  loop  in  an  oppo- 
site direction  to  what  they  did  before,  and  hence  the  induced  cur- 
rents which  result  from  the  increase  are  in  the  same  direction, 
referred  to  the  conductor,  as  during  the  first  quarter  revolution. 
During  the  next  two  quarter  revolutions  the  Induced  currents  will 
flow  in  an  opposite  direction.  Thus  by  continuous  revolution  the 
armature  is  traversed  by  currents  which  reverse  their  directions 
twice  each  revolution. 

In  order  to  lead  the  currents  from  the  armature  into  a  circuit 
where  they  can  be  used,  and  in  order  to  rectify  them,  i.e.,  cause 
them  to  flow  in  the  same  direction,  use  is  made  of  a  commutator. 
Fig.  391  represents  a  two-part  commutator  suited  for  our  single- 
loop  armature.  It  consists  in  an  insulating 
cylinder,  to  be  applied  to  the  extremity  of  the 
axis  of  the  armature.  Upon  it  is  slid  a  metal 
tube  slit  into  two  parts.  To  each  part  is  con- 
nected one  of  the  ends  of  the  loop,  as  shown  in 
Fig.  390.  Against  the  commutator  are  pressed 
two  spring  brushes,  B  B,  which  are  connected 
with  the  two  terminals  of  the  outside  circuit 
respectively.  The  commutator  revolves  with  the  armature,  but 
the  brushes  remain  stationary.  Both  are  so  arranged  that  at  the 
instant  the  plane  of  the  loop  of  the  armature  passes  through 
the  vertical  plane,  the  brushes  will  slide  from  one  segment  of  the 
commutator  to  the  other.  At  this  instant  the  induced  current 
reverses  the  direction  of  its  flow,  and  the  commutator,  exchanging 


FIG.  391. 


4:36  ELECTRICITY     AND     MAGNETISM. 

the  connections  with  the  external  circuit,  causes  the  external  cur- 
rent to  flow  in  one  direction. 

The  E.  M.  F.  which  could  be  obtained  from  such  an  ideal  dy- 
namo would  be  very  small.  To  increase  it,  the  total  number  of 
lines  of  force  which  are  passed  through  or  taken  out  of  the  circuit 
in  a  unit  time  must  be  increased.  There  are  three  ways  in  which 
this  may  be  accomplished :  the  speed  of  revolution,  the  number  of 
loops  in  the  armature,  or  the  strength  of  the  field  may  be  increased. 
It  need  not  be  considered  here  how  this  is  carried  out  in  practice. 

The  field  magnets  of  a  dynamo  may  be  excited  by  currents  from 
an  external  source,  by  the  whole  of  the  machine's  armature  current, 
or  by  only  a  portion  of  the  armature  current.  The  dynamos  are  then 
termed  separately  excited,  series,  or  shunt  machines  respectively. 

When  the  field  is  furnished  by  permanent  magnets  the  machine 
is  no  longer  termed  a  dynamo,  but  a  magneto-electrical  generator. 
Such  machines  are  not  a  commercial  success  except  in  the  very 
small  sizes. 

676.  Electric  Motors. — The  function  of  these  machines  is 
the  converse  of  that  of  dynamos.  They  are  intended  to  transform 
electrical  energy  into  motion.  The  dynamo  of  the  previous  article 
becomes  an  ideal  motor  by  simply  sending  through  it,  from  the  ex- 
ternal circuit,  a  current  in  an  opposite  direction.  The  commutator 
accomplishes  that  the  lines  of  force,  due  to  the  current  flowing  in 
the  armature,  shall  never  become  parallel  to  the  field's  lines.  In 
striving  to  secure  such  parallelism  the  armature  revolves  upon  its 
axis,  and  just  as  it  is  about  to  reach  the  goal  the  commutator  re- 
verses the  direction  of  its  lines,  and  it  moves  through  another  half 
revolution  to  be  again  frustrated  in  its  attempts. 


CHAPTER    IX. 

ELECTRO-CHEMISTRY    AND    ELECTRO-OPTICS. 

6T7.  Electrolytes. — Liquids  may  be  divided  into  three  classes, 
depending  upon  their  behavior  towards  the  electrical  current — 
those  which  do  not  conduct  at  all,  as  kerosene,  turpentine,  and  oils 
generally;  those  which  conduct  without  decomposition,  e.g.,  mer- 
cury and  molten  metals ;  those  which  are  decomposed  when  they 
conduct  a  current,  e.g.,  solutions  of  acids  or  metallic  salts  and  cer- 
tain fused  solid  compounds.  The  liquids  of  the  last  class  are 
called  electrolytes,  and  the  process  of  decomposing  an  electrolyte  by 


ELECTROLYSIS     OF     SALT     SOLUTIONS. 


437 


FIG.  392. 


means  of  an  electrical  current  is  termed  electrolysis.  The  two 
parts  into  which  the  electrolyte  is  decomposed  are  termed  ions. 

678.  Electrolysis   of   Sulphuric    Acid.  —  If   a  current  of 
electricity  flows  into  a  solution  of  sulphuric  acid  (H^SOJ  in  water 
by  means  of  an  electrode,  and  if,  after  traversing  the  solution,  it 
flows  out  through  another  electrode,  then  it  will,  by  its  passage, 
decompose  the  acid  into  two  parts — Ha  and  SO4.      The  hydrogen 
will  appear,  in  the  form  of  gas  bubbles,  at  the  electrode  through 
which  the  current  makes  its  exit  from  the  solution.     The  SO4  will 
endeavor  to  appear  at  the  electrode  where  the  current  entered, 
but  the  water  of  the  solution  seizes  upon  it,  and  together  they 
form  sulphuric  acid,  leaving,  however,  one  portion  of  oxygen  to 
appear,  as  gas,  at  the  electrode.     The  effect 

of  the  passage  of  the  current  is  to  virtually 
decompose  water  (H2O)  into  hydrogen  and 
oxygen,  there  being  twice  as  much  of  the 
former  as  of  the  latter. 

Hoffmann's  apparatus  for  electrolyzing 
sulphuric  acid  is  shown  in  Fig.  392.  The 
dilute  acid  solution  is  poured  into  the  funnel 
-F,  and  flowing  into  the  two  arms  of  the  front 
U-tube  fills  them,  providing  the  stop-cocks 
at  their  tops  be  opened.  After  filling,  the 
cocks  are  closed  and  a  current  is  made  to 
pass  between  the  two  platinum  electrodes  E 
E.  The  gases  which  are  evolved  at  the  elec- 
trodes rise  in  the  respective  tubes  above  them 
and  displace  the  liquid.  These  gases  are 
subjected  to  the  same  pressure  exerted  by 

the  liquid  in  the  funnel.  Their  volumes  may  be  read  off  from 
graduations  on  the  tubes  containing  them.  The  gases  may  be 
taken  off  through  the  cocks  and  their  natures  tested — the  oxygen 
being  made  to  relight  a  glowing  taper  and  the  hydrogen  being 
made  to  explode  when  mixed  with  air  in  a  test-tube. 

679.  Metallic  Salts.— When  the  electrolyte  is  a  metallic  salt 
solution  the  metal  will  be  deposited  at  the  electrode  where  the  current 
leaves  the  solution.     The  acid  of  the  salt  appears  at  the  other  elec- 
trode.    The  metal   may  be  deposited   upon   the   surface   of   tire 
electrode  in  the  form  of  a  thin  metallic  film.     In  case  the  metal 
has  a  strong  affinity  for  the  water  of  solution,  e.g.,  sodium  in  water, 
it  will  go  into  solution  and  hydrogen  will  be  evolved  as  a  secondary 
product.      It   is   nevertheless   true    that   Davy  obtained   metallic 
sodium  and  potassium  by  the  electrolysis  of  strong  caustic  soda 


438  ELECTRICITY     AND     MAGNETISM. 

and  potash.  These  metals  may  be  obtained  by  electrolysis,  if  a 
mercury  electrode  be  employed.  They  then  appear  in  the  form  of 
amalgams. 

The  character  of  a  deposited  metal  often  varies  under  different 
current  strengths  or  different  concentrations  of  solution.  Copper 
may  be  deposited  in  the  form  of  a  black  powder  instead  of  an 
even  metallic  film.  Silver  may  appear  in  the  form  of  crystals. 
Platinum  generally  appears  as  a  black,  finely  divided  sponge.  Tin, 
from  tin  chloride,  forms  a  beautiful  "tree"  of  tin  crystals,  the 
branches  spreading  out  gracefully  from  the  electrode. 

680.  Faraday's  Laws.  —  Faraday  proved  that  a  given  quan- 
tity of  electricity  always  deposits  the  same  weight  of  a  given  ion  from 
an  electrolyte  through  which  it  passes.     Thus  a  coulomb  of  elec- 
tricity always  deposits  .001118  gram  of  silver  on  an  electrode.     It 
makes  no  difference  whether  the  electrolyte  be  molten  silver  iodide 
or  chloride,  or  whether  it  be  a  water  solution  of  silver  nitrate,  sul- 
phate, acetate,  or  cyanide.     The  passage  of  one  coulomb  is  always 
accompanied  by  the  deposition  of  this  much  silver.     The  weights 
of  other  chemical  elements  which  a  coulomb  will  deposit  are  in  pro- 
portion to  their  chemical  equivalents    This  being  so,  it  must  be  con- 
cluded that  a  given  quantity  of  electricity  splits  up  the  same  number 
of  molecules,  whatever  the  electrolyte  may  be. 

681.  Voltameters.—  From  Faraday's  laws  it  will  be  readily 
seen  that  from  weighing  the  amount  of  an  ion,  which  is  deposited 
by  the  passage  of  a  certain  quantity  of  electricity,  this  quantity  may 
be  determined.     Thus,  if  a  certain  quantity  deposits  silver  on  an 
electrode  so  as  to  cause  it  to  weigh  1.118  gram  more  than  before 
the  passage,  it  is  evident  that  1,000  coulombs  have  passed.     If  the 
quantity  passed  in  the  form  of  a  constant  current,  which  lasted  for 
1  second,  then  the  current  strength  was  1,000  amperes.     For  an 
ampere  means  a  strength  of  current  which  delivers  1  coulomb  per 
second,  but  in  this  case  1,000  coulombs  were  delivered  in  a  second. 
In  general,  if  z  =  the  electro-chemical  equivalent  of  the  substance 
deposited,  i.e.,  grams  per  coulomb,  c  =  the  current  in  amperes, 
t  =  time  in  seconds  that  the  current  was  maintained,  the  weight 
of  the  substance  deposited* 

w  =  c  z  t. 

In  case  w  and  t  are  measured,  the  current  .strength  may  be  de- 
termined by  the  formula 


Instruments  for  measuring  current  strengths  in  this  manner  are 
called  voltameters.     The  substances  generally  employed  for  depo- 


ELECTROPLATING.  439 

sition  are  copper  from  a  solution  of  its  sulphate,  silver  from  its 
nitrate,  and  hydrogen  from  dilute  HaSO4.  In  the  case  of  hydro- 
gen weighing  is  difficult,  hence  the  volume  is  measured  and  then 
reduced  to  760  mm.  pressure  and  0°  C. 

A  current  of  1  ampere  deposits  in  1  minute,  of 

Hydrogen  (at  760  mm.  and  0   C.) 6.942  cu.  cm. 

Copper 01969  gram. 

Silver 06708  gram. 

Zinc 02018  gram. 

The  Edison  electrical  companies  place  zinc  voltameters  in  the         .  < 
houses  of  their  customers,  and  thus  measure  the  quantity  of  elec- 
tricity consumed. 

682.  Theory   of  Electrolysis. — The   most   satisfactory  ex- 
planation of  the  phenomena  of  electrolysis  is  embodied  in  the  the- 
ory of  Grotthuss,  somewhat  modified  by  Clausius.     The  molecules 
of  an  ordinary  solution  are  supposed  to  be  in  constant  vibration  in 
all  possible  directions.     Owing  to  collisions  between  the  molecules, 
or  other  causes,  the  constituent  atoms  are  constantly  leaving  their 
partners  and  combining  with  others  to  form  new  molecules.    Every 
molecule,  whatever  be  its  nature,  is  charged  with  the  same  quantity 
of  electricity — half  being  positive  and  half  negative.     The   positive 
resides  on  one  ion  of  the  molecule  and  the  negative  on  the  other. 
Now,  upon  subjecting  the  solution  to  a  difference  of  potential  be- 
tween  the  electrodes,  the  direction   of   the  molecular  motions  is 
controlled,   and  ions,  which  by  chance  are  isolated,  will  teud  to 
move  towards  one  or  the  other  electrode,  according  to  the  signs 
of  the  charges  which  are  upon   them.     If  the  impressed  electro- 
motive force  is  large  enough  to  prevent  recombination  of  these  ions, 
they  will  continue  their  movements  towards  the  electrodes,  and 
will  accumulate  around  them.     Upon  touching  the  electrodes  they 
impart  to  them  their  minute  charges  and  the  continuous  accumula- 
tion of  these  maintains  a  current  in  the  circuit. 

According  to  this  theory,  electrolytic  conduction  of  electricity 
is  similar  to  the  convection  of  heat  in  liquids.  The  transportation 
of  electricity  is  accompanied  by  a  transportation  of  matter. 

The  remarkable  connection  between  the  results  of  the  quanti- 
tative woi-k  of  Faraday  and  the  chemical  equivalents  of  the  ele- 
ments, points  to  electrolysis  as  a  fertile  field  for  the  investigation 
of  the  yet  unknown  nature  of  chemical  affinity. 

683.  Electroplating. — The  principles  of  electrolysis  are  made 
use  of  in  the  mechanic  arts.     Articles  made  of  baser   metals  are 
covered  over  with  a  thin  deposit  of  silver  or  gold  and  are  said  to 


440  ELECTRICITY     AND     MAGNETISM. 

have  been  electroplated.  The  articles  to  be  plated  are  suspended 
in  a  bath  from  a  metallic  rod,  which  is  in  electrical  communication 
with  the  negative  pole  of  a  battery  or  dynamo  (Fig.  393).  The 

bath  consists  of  a  solution 

Fl°-  :5J):'-  of   some  salt   of  the  metal 

which  is  to  be  deposited, 
e.g.,  silver  or  gold  cyanide. 
The  current  from  the  posi- 
tive pole  of  the  dynamo  en- 
ters the  solution  by  means 
of  an  electrode,  C,  made  of 
the  same  metal  as  that  which 
is  contained  in  the  salt. 
Upon  passing  a  current,  the 
salt  of  the  solution  is  decomposed — the  metal  depositing  on  the 
article  to  be  plated,  and  the  acid  combining  with  the  electrode  C 
to  form  new  salt,  thus  maintaining  the  concentration  of  the  solu- 
tion. The  articles  to  be  plated  must  be  thoroughly  scoured  and 
cleansed  before  immersion  in  the  bath.  The  character  of  the 
results  obtained  depends  much  upon  the  character  and  concen- 
tration of  the  baths  and  upon  the  magnitude  of  the  currents  and 
electro-motive  forces  employed.  Full  details  must  be  looked  for 
in  technical  books. 

684.  Electrotyping. — If  the  object  to  be  plated  consists  of 
an  impression,  in  wax  or  paper  pulp,  of  the  type  from  which  a 
page  is  printed,  the  impression  having  been  coated  with  fine  plum- 
bago to  render  it  a  good  conductor,  copper  deposited  upon  it  may 
be  removed,  and  having  been  stiffened  by  melted  lead  (or  some 
alloy)  poured  over  its  under  surface,  it  may  be  used  in  the  print- 
ing-press instead    of   the  type.     It  is  then   called  an  electrotype 
phtti',  and  when  not  in  use  may  be  preserved  indefinitely  for  suc- 
ceeding editions,   while  the  type  of  which  it  is  a  copy  can  be 
distributed  and  used  for  other  purposes. 

685.  Counter-Electromotive  Force. — If  a  current  be  sent 
through  a  solution  of  alkaline  zincate  by  means  of  two  copper 
electrodes,  zinc  will  be  deposited  on  one  electrode  and  the  other 
will  become  oxidized.     If  the  connections  with  the  source  of  elec- 
tricity be  now  removed  and  transferred  to  an  electric  bell,  the  bell 
will  ring.     The  bath  and  electrodes  have  been  transformed  into  a 
galvanic  cell.    The  current  which  it  gives  is  in  a  direction  opposite  to 
that  which  caused  the  decomposition  of  the  solution.     Its  E.  M.  F. 
is  about  0. 79  volt,  and  is  opposed,  to  the  original  E.  M.  F.     Had 
the  original  E.  M.  F.  been  less  than  this  amount,  no  plating  of  the 


STORAGE     BATTERIES.  44J 

electrodes  could  have  occurred.  The  E.  M.  F.  developed  in  the 
solution  is  termed  a  counter-electromotive  force.  It  occurs  in  nearly 
all  electrolytic  actions,  except  when  the  electrodes  are  of  the  same 
metal  as  that  which  is  being  deposited,  e.g.,  copper  in  copper 
sulphate. 

The  counter-electromotive  force  developed  in  the  electrolysis 
of  dilute  sulphuric  .acid  is  about  1.47  volt.  Hence,  to  perform 
the  electrolysis,  more  than  one  Daniell's  cell  is  necessary. 

The  counter-electromotive  force  constitutes  the  polarization  of 
a  primary  battery  mentioned  in  Art.  634. 

686.  Storage  Batteries. — The  copper  electrodes  in  alkaline 
zincate  of  the  preceding  article  represent  a  very  simple  form  of 
storage  battery  or  electrical  accumulator.  Upon  sending  a  current 
through  it,  the  zinc  is  deposited  and  the  battery  is  said  to  be 
charged.  Some  of  the  electrical  energy  has  been  transformed. into 
chemical  energy.  The  electrbdes  may  be  removed  from  the  solu- 
tion, packed  away,  and  then  be  brought  forward  in  the  future  and 
be  made  to  turn  back  their  energy  into  electricity.  The  elec- 
tricity proper  has  not  been  stored  away,  but  the  energy  repre- 
sented by  it. 

The  first  successful  storage  battery  was  constructed  by  Gaston 
Plante  in  1860.  His  electrodes  were  made  of  sheet-lead,  and  the 
electrolyte  was  dilute  sulphuric  acid.  In  order  to  expose  a  large 
surface  of  electrodes  he  made  them  of  large  sheets  which  he  coiled 
up  into  spirals,  as  shown  in  Fig.  394,  the  two  plates  being  insulated 
from  each  other  by  rubber  bands  between  the  spirals. 
The  object  of  the  large  surface  was  to  increase  the 
capacity  of  the  cell.  The  spiral  form  was  conducive 
to  a  small  internal  resistance.  Upon  the  passage  of  a 
current  the  acid  was  decomposed  and  hydrogen  re- 
duced one  electrode  to  bright  metallic  lead,  while 
oxygen  coated  the  other  with  peroxide  of  lead. 
These  two  conditions  of  the  electrodes  rendered 
them  capable  of  giving  an  electro-motive  force  of  two 
volts.  By  repeated  charging  in  alternate  directions 
the  surfaces  of  both  electrodes  were  rendered  spongy, 
thus  exposing  an  increased  surface  to  the  action  of 
the  ions  and  increasing  the  capacity  of  the  cell  accordingly.  This 
preliminary  alternate  charging  was  termed  by  him  "formation"  of 
the  electrodes,  and  was  performed  at,  the  expense  of  costly  currents. 

In  order  to  reduce  the  time  and  expense  of  formation,  Faure 
used  lead  plates  as  a  support  and  covered  them  with  a  paste 
made  of  powdered  oxide  of  lead  mixed  with  sulphuric  acid.  This 
paste  he  kept  in  place  by  covering  the  sheets  with  felt.  When  the 


442 


ELECTRICITY     AND     MAGNETISM 


FIG.  395. 


charging  current  was  connected  the  oxide  on  one  plate  was  changed 
to  a  higher  oxide,  and  on  the  other  plate  transformed  into  metallic 
sponge.  This  idea  of  Faure  was  an  excellent  one,  and  is  at  the 
foundation  of  the  construction  of  all  the  commercial  lead  accu- 
mulators. The  percentage  of  energy  recovered  by  discharge  wa» 
greatly  increased.  His  method  of  keeping  the  paste  in  place  by 
felts  was,  however,  soon  abandoned,  because  fine  lead  needles  soon 
filled  up  the  interstices  of  the  felt,  and  thus  made  a  metallic  con- 
nection between  the  electrodes.  Holes  were  then  punched  in  the 
lead  plates  and  the  paste  pressed  into  them.  A  large  number  of 
the  patents  recently  issued  for  accumulators  refer  to  methods  of 
making  these  holes  and  pressing  in  the  paste,  or  to  the  shape  of  the 
holes  themselves  after  they  have  been  punched.  The  shapes  vary 
from  a  slight  depression  on  the  surface  to  a  hole  completely 
through  the  plate,  and  even  further,  to  a  hollow  plate,  with  small 

openings  leading  to  the  surface. 
A  great  deal  depends  upon  this 
shape,  for  the  paste  changes  its 
volume  during  the  process  of 
charging  and  discharging,  and  it 
would  tend  to  loosen  itself  from 
some  shaped  openings  and  fall 
to  the  bottom  of  the  cell,  while 
in  others  it  would  tend  to  tighten 
itself,  and  thus  provide  a  better 
contact. 

A  modern  commercial  stor- 
age battery  is  shown  in  Fig.  395. 
The  electrodes  are  made  up  of  a 
number  of  pasted  plates,  or  grids 
as  they  are  called.  The  grids  of 

one  electrode  are  alternated  with  those  of  the  other,  and  are  all 
connected  by  lugs  with  com- 
mon cross-bars  which  consti- 
tute the  terminals  of  cell. 


FIG.  396. 


687.  Capillary  Elec- 
trometer.— This  instrument 
is  for  the  measurement  of 
small  differences  of  potential 
not  exceeding  1  volt.  A  simple 
form  is  represented  in  Fig. 
396.  It  consists  of  two  up- 
right test-tubes  connected  by  a  horizontal  capillary  glass  tube  of 
about  £  mm.  internal  diameter. 


LIGHT     AND     ELECTRICITY.  44£ 

Into  one  of  the  test-tubes  is  poured  mercury  and  into  the  other 
dilute  sulphuric  acid.  The  heights  of  the  two  liquids  are  so  arranged 
that  the  dividing  surface  between  them  shall  be  in  the  horizontal  tube, 
Upon  subjecting  the  two  liquids  to  an  electro-motive  force,  applied 
at  two  platinum  terminals  fused  into  the  bottoms  of  the  test-tubes,, 
an  electrolytic  action  will  be  started  at  the  point  of  the  capillary 
tube  where  the  acid  meets  the  mercury.  The  surface  tension  will 
be  accordingly  modified  and  the  balance  between  the  two  columns 
will  be  destroyed.  To  reproduce  a  balance  the  dividing  surface 
must  move  along  the  capillary  tube  in  one  direction  or  the  other, 
depending  upon  which  liquid  has  the  higher  potential.  The  dis- 
tance moved  depends  upon  the  potential  difference  and  becomes 
a  measure  of  it. 

688.  Light  and  Electricity. — At  the  present  time  many  in- 
vestigators are  experimenting  upon  the  close  relation  between  the 
phenomena  of  light  and  those  of  electricity.  Trustworthy  results 
point  to  the  fact  that  electricity  is  the  lurniniferous  ether  itself,  as 
was  previously  stated.  A  motion  of  the  ether  is  unrestrained  in  a 
perfect  electrical  conductor.  In  a  dielectric  only  a  limited  dis- 
placement of  the  ether  particles  is  possible,  except  in  case  the 
dielectric  is  ruptured.  A  displacement  always  subjects  the  enclos- 
ing dielectric  to  a  strain,  and  can  be  produced  by  a  neighboring 
conductor  having  an  electrostatic  charge  or  by  its  conveying  an 
electrical  current.  The  displacement  resulting  from  a  current  is 
in  a  direction  opposite  to  the  current,  and  occurs  through  all  the 
dielectric  which  surrounds  the  current.  Upon  starting  the  current 
the  displacements  near  the  conductor  occur  before  those  at  a  dis- 
tance. The  velocity  of  propagation  of  the  first  impulse  causing 
displacement  is  the  same  as  the  velocity  of  light. 

A  full  exposition  of  the  ether  hypothesis,  and  to  what  extent 
it  explains  electrical  phenomena  is,  of  course,  out  of  place  here. 

639.  Double  Refraction  from  Electrostatic  Strain.  — 
Kerr  showed  that  the  strain  in  a  dielectric,  caused  by  electrostatic 
difference  of  potential,  could  be  detected  by  means  of  polarized 
light.  He  placed  a  block  of  glass  between  two  Nicol's  prisms 
which  served  as  analyzer  and  polarizer.  Into  opposite  sides  of  the 
glass  were  bored  two  holes,  not  quite  meeting  each  other,  but  sep- 
arated by  about  2  mm.  Into  these  holes  were  placed  wires,  which 
were  connected  with  the  poles  of  a  Holtz  machine.  Upon  creating 
a  difference  of  potential  between  the  ends  of  the  wires  the  glass 
was  subjected  to  strain  and  exhibited  to  an  eye  placed  at  the  an- 
alyzing Nicol  similar  colors  to  those  given  by  mechanically  strained 
glass.  The  glass  was  made  doubly  refracting. 


444  ELECTRICITY     AND     MAGNETISM. 

690.  Magneto-Optic  Twisting  of  the  Plane  of  Polarized 
Light. — Faraday  discovered  that  the  plane  of  polarization  of  a  ray 
of  light  which  traversed  a  magnetic  field  in  a  direction  parallel  to 
the  lines  of  force  was  twisted  by  the  field.     One  form  of  Faraday's 
experiment  is  to  place  a  straight  electro-magnet  between  two  Nic- 
ol's  prisms,  which  have  been  crossed  so  as  to  produce  extinction  of 
light.     Substitute  for  the  iron  core  of  the  magnet  a  tube  with 
glass  ends,  which  is  filled  with  bisulphide  of  carbon.     Before  the 
magnet  is  excited  a  ray  of  light  from  the  polarizer  passes  through 
the  liquid  and  is  brought  to  extinction  by  the  analyzer.     If,  now, 
the  magnet  be  excited  by  an  electrical  current,  the  analyzer  no 
longer  extinguishes  the  ray,  and  that  it  may  do  so  must  be  rotated 
through  a  certain  angle.     The  plane  of  the  ray  has  been  twisted 
or  rotated  by  the  magnetic  field.     The  direction  of  the  rotation  is 
the  same  as  the  direction  of  the  exciting  current.     By  reversing 
the  current  the  plane  will  be  twisted  in  an  opposite  direction.   The 
amount  of  the  rotation  of  the  analyzer  necessary  to  reproduce  ex- 
tinction of  the  ray  is  directly  proportional  to  the  length  of  the 
tube  and  to  the  strength  of  the  magnetic  field,  i.e.,  to  the  strength 
of  the  exciting  current.     It  also  depends  upon  the  nature  of  the 
liquid  in  the  tube.     In  general  it  may  be  said  that  substances  of 
high  refractive  indices  have  large  rotatory  powers. 

As  might  be  expected,  ra}*s  of  the  different  colors  are  rotated 
through  different  angles.  Hence,  if  complete  extinction  by  large 
rotations  be  desired,  monochromatic  light  should  be  used. 

691.  Rotation  of  the  Plane  by  Reflection.— Kerr  discov- 
ered that  the  plane  of  polarization  was  rotated  when  the  ray  was 
reflected  from  the  polished  pole  of  the  iron  core  of  an  electro-mag- 
net.    In  this  case  the  direction  of  rotation  was  contrary  to  the 
direction  of  the  magnetizing  currents. 

692.  Photo-Electric  Properties  of  Selenium.— Selenium, 
when  thoroughly  annealed,  offers  a  resistance  to  an  electric  current 
which  is  dependent  upon  the  degree  to  which  it  is  illuminated. 
An  increase  of  illumination  decreases  the  resistance.     A  piece  of 
selenium,  whose  resistance  in  the  dark  was  500  ohms,  has  been 
known  to  decrease  its  resistance  to  50  ohms  upon  exposure  to 
bi'ight  sunlight. 

This  peculiarity  of  selenium  is  made  use  of  by  Bell  in  the 
construction  of  his  photophone.  This  instrument  is  intended  for 
transmitting  sounds  to  a  distance  by  means  of  rays  of  light,  which 
are  reflected  from  a  mirror  that  is  made  to  vibrate  by  the  sounds. 
Light  of  varying  intensity  is  made  thus  to  impinge  upon  a  piece  of 
selenium,  which  is  connected  in  circuit  with  a  battery  and  a  Bell 


POWER     OF     ELECTRICAL     CURRENTS.  445 

telephone  receiver.  The  variations  in  the  resistance  of  the  selen- 
ium, because  of  the  varied  illumination,  cause  variations  of  the  cur- 
rent in  the  receiver,  which  serve  to  reproduce  the  sounds. 

Quite  recently  Shelford-Bidwell  has  exhibited  an  apparatus  in 
which  selenium  is  made  light  the  gas  as  darkness  comes  on  and  to 
turn  it  off  as  daylight  appears. 

Problems. 

1.  How  much  copper  will  be  deposited   by    a    current  of    3     £ 
amperes  in  an  hour  ? 

"   2.  A  current  of  0.5  ampere  is  used  for  preparing  pure  silver  OvX 
by  electrolysis :  how  long  must  the  current  be  allowed  to  flow  in 
order  to  obtain  a  deposit  of  4  grams  ? 

'    3.  What  is  the  strength  of  a  current  which  deposits  a  milli-    ^ 
gram  of  copper  per  minute  ? 

4.  It  is  found  that  a  current  of  1.868  ampere  deposits  1.108 
gram  of  copper  in  half  an  hour  :  what  value  does  this  give  for  the 
electro-chemical  equivalent  of  copper  ? 

5.  What  is  the  strength  of  a  current  which  deposits  0.935  gram 
of  copper  in  1  hour  and  10  minutes. 


CHAPTER    X. 

THE  RELATIONS  BETWEEN  ELECTRICITY  AND  HEAT. 

693.  Power  of  the  Electrical  Current.  —  A  current  whose 
strength  is  c  carries  in  t  seconds  c  t  units  of  electricity  from  a 
potential  Fto  one  of  V.  The  work  which  has  to  be  expended  in 
doing  this  is  c  t  (V  —  V),  as  was  shown  in  Art.  567.  In  this  case 
V  —  V  is  equal  to  the  electro-motive  force  E,  which  is  sending  the 
current.  Hence,  representing  the  work  by  A,  we  have 

A  —  c  t  E. 

If  c,  t,  and  E  are  measured  in  absolute  units,  the  work  is  given  in 
ergs. 

The  power  of  the  current  P  being  the  rate  at  which  the  work 
is  done,  i.e.,  the  work  divided  by  the  time  required  to  perform  it 
is  expressed  by  the  formula 


Expressing  c  and  E  in  amperes   and  volts  respectively  will   di- 


446  ELECTRICITY    AND    MAGNETISM. 

vide  the  ergs  per  second  by  107.     This  gives  the  power  in  watts 
(Art.  38). 

Inasmuch  as  c  =  —  and  E  =  c  R,  by  Ohm's  law,  these  values 
may  be  substituted,  and  we  have,  further, 


p  =  e  E. 

694.  Heat   Developed  in  a  Conductor.  —  Whenever  the 
energy  which  is  represented   by  a  current  is  not  expended   in 
doing  external  work,  as  in  driving  motors  or  decomposing  electro- 
lytes, it  is  transformed  into  heat.     The  conductor  which  carries 
the  current  becomes  heated.     If  a  conductor  of  resistance,  E,  car- 
ries a  current  c,  then,  by  Ohm's  law,  the  difference  of  potential 
between  its  ends,  E  =  c  R.     The  energy  represented  by  the  cur- 
rent is,  as  in  the  preceding  article, 

A  =  c  IE  =  c*  E  t  ergs. 

This  energy  is  transformed  into  heat.  To  express  the  heat  in 
gram-calories,  Joule's  mechanical  equivalent  of  heat  must  be  intro- 
duced. Without  going  through  with  the  transformations  it  is 
sufficient  to  say  that  a  current  of  c  amperes  flowing  for  t  seconds 
through  E  ohms  communicates  to  the  conductor  carrying  it 
H  =  c*  E  t  0.24  gram-calories. 

695.  Rise  in  Temperature  of  the  Conductor.  —  A  long 
thick  wire  could  have  the  same  resistance  as  a  short  thin   one,  but 
a  given  current  traversing  them  for  a  given  time  would  produce 
the  same  quantities  of  heat  in  each.     The   short  thin  wire,   not 
weighing  so  much,  might  have  its  temperature  raised  several  hun- 
dred degrees,  while  the  thick  wire  would  suffer  a  rise  of  a  few  de- 
grees only. 

In  order  to  determine  what  rise  in  temperature  will  accompany 
a  given  quantity  of  heat  imparted,  account  must  be  taken  of  the 
dimensions  of  the  conductor,  the  specific  heat  of  the  substance  of 
which  the  conductor  is  composed,  and  the  temperature  coefficient 
of  the  conductor,  i  P.  ,  the  amount  by  which  its  resistance  would  in- 
crease under  a  rise  of  one  degree  of  temperature.  A  full  considera- 
tion cannot  be  considered  in  these  chapters.  It  is  well  to  know, 
however,  that,  in  different  wires  of  the  same  material,  traversed  by  the 
same  current,  the  rise  in  temperature  is  inversely  proportional  to  the 
fourth  power  of  their  diameters. 

A  wire  of  given  resistance,  traversed  by  a  given  constant  current, 
will  receive  the  same  amount  of  heat  each  second  that  the  current 
flows.  After  a  short  time  the  temperature  of  the  wire  may  rise  to 


HOT    WIRE     AMMETERS.  447 

such  a  point  that  it  gives  off  to  surrounding  objects,  by  radiation 
and  conduction,  just  as  much  heat  as  it  receives  in  every  second. 
The  temperature  then  remains  constant  at  this  point  as  long  as 
the  flow  is  maintained. 

The  heat  effects  mentioned  may  be  illustrated  by  sending  a 
strong  current  through  a  chain,  whose  alternate  links  are  made  of 
platinum  and  silver  wire.  The  platinum  links  will  be  heated  to 
luminosity  while  the  appearance  of  the  silver  remains  unaltered. 
The  reason  for  this  is  that  the  platinum  offers  a  much  greater  re- 
sistance than  the  silver,  and  its  specific  heat  is  less. 

Platinum  wires,  heated  red-hot  by  currents,  are  much  used  by 
surgeons  for  cauterization.  They  are  much  easier  of  manipulation 
than  the  knife. 

696.  Hot  Wire  Ammeters  and  Voltmeters.— The  expan- 
sion in  length  which  a  wire  undergoes  when  its  temperature  is 
raised  to  a  certain  point  by  a  current  which  traverses  it,  can  be  made 
a  measure  of  the  strength  of  the  current.  A  given  wire  has  a 
definite  length  at  a  given  temperature.  Increasing  the  tempera- 
ture increases  the  length.  Every  current  produces  a  definite  length 
in  the  wire.  Different  current  strengths  correspond  to  different 
lengths.  A  measurement  of  the  length  can  thus  be  made  a  meas- 
ure of  the  current  strength. 

A  simple  ammeter,  whose  action  depends  upon  this  principle,  is 
represented  in  Fig.  397.  The  current  to  be  measured  is  passed 

FIG.  397. 


1 


through  a  long  and  thin  platinum  or  iron  wire,  one  of  whose  ends 
is  clamped  in  a  stationary  binding-post.  The  other  end  passes 
around  and  is  fastened  to  a  small  metallic  cylinder.  This  cylinder 
turns  upon  a  metallic  pivot  fastened  in  another  binding-post. 
The  current  having  traversed  the  wire  leaves  it  by  this  binding-post. 
The  wire  is  subjected  to  a  constant  strain,  exerted  by  a  spiral  spring 
attached  to  the  periphery  of  a  disc,  which  is  fastened  to  one  end 
of  the  cylinder.  The  disc  carries  a  radial  pointer,  whose  end 
moves  over  a  graduated  scale  whenever  the  length  of  the  wire  is 
changed  by  a  change  in  temperature  caused  by  a  current.  The 


448  ELECTRICITY     AND     MAGNETISM. 

graduation  of  the  scale  is  empirical,  being  determined  by  the  assist- 
ance of  some  other  current  measurer. 

As  the  current  strength  is  dependent  upon  the  difference  of  po- 
tential between  the  two  binding-posts,  it  is  evident  that  the  instru- 
ment may  be  graduated  as  ^voltmeter,  i.e.,  will  indicate  the  volts  im- 
pressed upon  it.  As  it  is  not  desirable  that  a  large  current  should 
flow  through  a  voltmeter,  the  wire  of  such  an  instrument  should  have 
a  large  resistance.  The  voltmeters  of  Cardew  are  constructed  on 
this  principle.  Sometimes  a  high  resistance  coil  is  inserted  in 
series  with  the  wire,  and  then  the  voltmeter  readings  indicate  the 
fall  in  potential  between  the  terminals  of  the  spool  and  wire  in 
series. 

Hot  wire  ammeters  and  voltmeters  can  be  employed  to  meas- 
ure currents  and  voltages  which  rapidly  alternate  their  directions. 
For  the  heat  produced  being  dependent  on  the  square  of  the  cur- 
rent strength  is  positive,  whether  the  current  flows  in  a  positive  or 
negative  direction. 

697.  Electric  Welding.  —  The  welding  together  of  two 
pieces  of  metal,  by  means  of  the  electric  current,  as  done  in  the 

Thomson  process,  depends  upon  the  heat 
j?  IG.  oyo*  . 

produced.  The  pieces  are  pressed  to- 
gether and  a  powerful  current  (some- 
times 50,000  amperes)  is  sent  across  the 
juncture.  The  consequent  heat  renders 
the  metal  plastic,  and  upon  cooling  a 
most  perfect  joint  is  obtained. 

698.  The  Electric  Arc. — If  two 
rods  of  carbon,  traversed  by  a  current 
from  a  source  of  at  least  40  volts  electro- 
motive force,  be  touched  together  ut  their 
ends  and  then  be  separated  by  a  few 
millimeters'  distance,  an  electric  flame  or 
arc  will  be  observed  to  pass  over  this  dis- 
tance. A  brilliant  light  will  accompany 
it,  the  extreme  brilliancy  being  at  the 
end  surfaces  of  the  rods.  If  allowed  to 
burn  for  a  few  moments  the  rods  and 
flame  will  present  an  appearance  like  that 
represented  in  Fig.  398.  The  end  of  the 
positive  rod  will  have  formed  itself  into  a 
sort  of  crater,  while  the  end  of  the  neg- 
ative will  have  become  pointed.  If  al- 
lowed to  burn  for  some  time,  the  rods  will  be  consumed,  and,  in  a 


THERMO-ELECTRICITY.  449 

given  time,  about  twice  as  much  of  the  positive  rod  will  be  con- 
sumed as  of  the  negative. 

In  order  to  form  an  arc  it  is  necessary  that  the  points  be  at 
first  in  contact.  When  in  loose  contact  the  current  encounters  a 
great  resistance,  and  accordingly  heats  the  points  until  a  tempera- 
ture is  reached  which  is  sufficient  to  vaporize  the  carbon.  Carbon 
vapor  is  a  nmch  better  conductor  of  electricity  than  air,  and 
whereas  an  arc  could  not  be  maintained  across  an  air  space,  yet  it 
can  be  across  a  space  filled  with  this  vapor. 

The  heat  at  the  vapor  portion  of  the  arc  is  intense,  being  suffi- 
cient to  vaporize  the  most  refractory  substances,  of  which  carbon 
itself  is  the  best  example.  The  heat  at  the  crater,  though  not  so 
intense,  is  the  cause  of  greater  illumination,  because  of  being  asso- 
ciated with  a  solid  instead  of  a  vapor. 

Eecent  investigations,  concerning  the  fall  of  potential  along  the 
arc,  indicate  that  a  large  portion  of  the  electrical  energy  repre- 
sented by  it  is  consumed  in  maintaining  the  heat  of  the  crater. 

699.  Incandescent  Electric    Lamps.— These    lamps   con- 
sist of  filaments  of  carbonized  bamboo,  paper,  or  silk,  which  are 
heated  to  incandescence  by  the  current.     That  the  filaments  may 
not  be  consumed  by  combustion,  they  are  sealed  into  glass  bulbs, 
from  which  the  air  has  been  exhausted.     Although  no  oxygen  is 
present,  the  filaments  become  disintegrated  by    continuous   use. 
Particles  of  carbon  escape  from  the  surface  of  the  filament  and  are 
oftentimes  deposited  upon  the  interior  of  the  bulb,    causing   a 
brownish  opalescent  appearance. 

700.  Thermo-Electricity.—  Let  two  bars  of  bismuth  (6)  and 
antimony  (a)   be  soldered  together  as  in  Fig.  399.     If,  now,  the 
joint  S  be  heated   by  a 

lamp  a  current  will  flow  FlG-  a 

across  the  heated  June- 
tion  from  the  bismuth  to 
the  antimony,  as  will  be 
shown  by  the  galvano- 
meter G. 

The  electro-motive  force  of  the  current  depends  upon  the  metals 
in  contact  at  the  heated  junction.  If  any  one  of  the  metals  given 
below  be  joined  with  any  one  following  it  in  the  list,  upon  apply- 
ing heat  the  current  will  flow  across  the  junction  from  the  former 
to  the  latter :  Bismuth,  lead,  platinum,  tin,  zinc,  copper,  iron,  an- 
timony. 

The  thermo-electro-motive  force  is  proportional  to  the  difference  of 
temperature  between  UK:  junction  and  the  rest  of  the  circuit. 


450 


ELECTRICITY    AND     MAGNETISM. 


The  E.  M.  F.  of  a  single  thermoelement  is  very  small.  If  the 
junction  of  a  copper-iron  element  be  heated  1°  C.  above  the  tem- 
perature of  the  rest  of  the  circuit,  the  E.  M.  F.  developed  is  about 
fourteen  millionths  of  a  volt. 

In  some  cases,  e.g.,  with  iron,  a  continued  increase  of  tempera- 
ture at  the  junction  finally  reverses  the  direction  of  the  current. 

TOl.  Thermo-Electric  Pile. — If  a  series  of  bars  of  bismuth 
and  antimony  be  arranged,  as  in  Fig.  400,  and  the  junctions 
marked  3  and  4  be  equally  heated, 
no  current  will  be  indicated  by  the 
galvanometer ;  for  the  flow  at  3 
would  be  from  the  bismuth  to  the 
antimony  as  indicated  by  the  arrow, 
while  at  4  it  would  also  be  from  b 
to  a,  as  shown,  and  these  two  cur- 
rents would  neutralize  each  other. 
But  if  we  heat  only  one  set  of  junctions,  the  odd-numbered  for 
instance,  then  a  current  flows  whose  electro-motive  force  is  pro- 
portional to  the  number  of  heated  junctions. 

A  set  of  twenty  or  thirty  pairs,  conveniently  arranged  so  that 
the  alternate  junctions  may  be  simultaneously  subjected  to  heating 
or  cooling  effects,  is  called  a  thermo-pile,  and  has  been  an  impor- 
tant instrument  in  investigations  upon  radiant  heat. 

702.  Peltier  Effect. — Peltier  discovered  a  phenomenon  which 
is  the  converse  of  that  mentioned  in  the  preceding  articles.  He 
found  that,  if  a  current  of  electricity  be  sent  through  a  junction  of 
dissimilar  metals,  the  junction  becomes  heated  or  cooled  according 
to  the  direction  of  the  current.  For  instance,  if  a  current  be  sent 
through  a  junction  from  bismuth  to  antimony,  the  junction  will 
absorb  heat,  i.e.,  become  cooled.  If  the  current  be  reversed  the 
junction  will  become  heated. 

The  heat  thus  produced  is  not  owing  to  the  resistance  of  the 
conductors.  For  the  heat  from  resistance  is  not  altered  by  a 
change  in  the  direction  of  the  flow  of  the  current.  Cooling  can 
never  result  from  ohmic  resistance.  Again,  the  heat  of  the  Peltier 
effect  is  proportional  to  the  current  strength  simply,  whereas  the 
heat  from  resistance  is  pi-oportional  to  the  square  of  the  current 
strength. 

Problems. 

1.  An  11,000  watt  dynamo  develops  an  E.  M.  F.  of  110  volts : 
{a}  What  is  the  current  strength  in  the  mains  ?  (b)  How  many  in- 
candescent lamps,  of  220  ohms  hot  resistance,  will  it  light,  provid- 


THE     ELECTBICAL     UNITS.  451 

ing  they  are  arranged  in  multiple  arc  ?  (c)  How  many  gram-calo- 
ries will  be  developed  in  each  lamp  per  second  ?  (d)  How  many 
•watts  will  be  consumed  by  each  lamp  ?  [  (a)  100  amperes. 

A         I  (6)  200  lamps. 
'**  1  (c)  13.2  calories. 
L  (d)  55  watts. 

2.  How  much  power  is  required  to  properly  operate  an  arc 
lamp  which  carries  10  amperes  and  has  a  difference  of  potential  of 
45.2  volts  between  its  terminals? 

3.  How  many  calories  are  developed  per  minute  in  a  wire  of 
100  ohms  resistance,  traversed  by  5  amperes  ? 

4.  A  wire  of  2  ohms  resistance  placed  in  100  grams  of  water  is 
traversed  by  a  certain   current,  which,  in  20  minutes,  raises  the 
temperature  of  the  water  from  18°  to  28°  C.:  what  is  the  current 
strength?  A n*.  1.32  amperes,  nearly. 

The  Electrical  Units. 

Electrical  magnitudes  may  be  expressed  in  three  different  sets 
of  units.  Two  of  them — the  absolute  electrostatic  and  the  absolute 
electro-magnetic  units — are  termed  absolute  because  they  are  units 
derived  from  the  absolute  units  (Art.  4)  of  length,  mass,  and  time, 
viz.,  the  centimetre,  gram,  and  second.  The  third  set  are  called 
practical  units,  because  they  are  the  ones  which  are  employed  by 
practical  electricians.  They  are  either  decimal  multiples  or  deci- 
mal parts  of  the  electro-magnetic  units. 

ELECTROSTATIC  UNITS. 

The  Unit  of  Quantity  of  electricity  is  that  quantity  which,  when 
placed  at  a  distance  of  one  centimetre  from  a  similar  and  equal 
quantity,  repels  it  with  a  force  of  one  dyne  (Art.  5G3). 

The  Unit  Strength  of  Current  flows  in  a  circuit  when  a  unit 
quantity  of  electricity  passes  any  section  of  the  conductor  in  one 
second. 

The  Unit  Difference  of  Potential  exists  between  two  points 
when  it  requires  an  expenditure  of  one  erg  of  work  to  bring  a 
unit  quantity  of  electricity  from  one  point  to  the  other  against  the 
electric  force. 

The  Unit  of  Resistance  is  offered  by  that  conductor  which,  when 
interposed  between  two  bodies  whose  potentials  are  maintained  at  a 
constant  difference  of  unity,  allows  a  unit  current  to  pass  along  it. 

The  Unit  of  Capacity  is  possessed  by  that  conductor  which 
requires  that  it  be  charged  with  a  unit  quantity  of  electricity  in 
order  that  its  potential  may  be  raised  from  zero  to  unity. 

ELECTRO-MAGNETIC  UNITS. 
The  Unit  Strength  of  Current  is  such  that,  when  flowing  through 


452  ELECTRICITY     AND     MAGNETISM. 

a  conductor  of  one  centimetre  length  which  is  bent  into  an  arc  of 
one  centimetre  radius,  it  will  exert  a  force  of  one  dyne  on  a  unit 
magnetic  pole  situated  at  the  centre. 

The  Unit  Quantity  of  electricity  passes  in  one  second  through  a  sec- 
tion of  a  conductor  which  is  traversed  by  a  current  of  unit  strength. 

The  Unit  Difference  of  Potential  (or  of  Electro-motive  Force)  exists 
between  two  points  when  it  requires  the  expenditure  of  one  erg  of 
work  to  bring  a  unit  of  electricity  from  one  point  to  the  other 
against  the  electric  force. 

The  Unit  of  Resistance  is  offered  by  that  conductor  which,  when 
interposed  between  two  bodies  whose  potentials  are  maintained  at 
a  constant  difference  of  unity,  allows  a  unit  current  to  pass  along  it. 

The  Unit  of  Capacity  is  possessed  by  that  conductor  which 
requires  that  it  be  charged  with  a  unit  quantity  of  electricity  in 
order  that  its  potential  may  be  raised  from  zero  to  unity. 

A  little  consideration  will  show  that  in  the  electrostatic  and 
electro-magnetic  systems  the  definitions  of  all  the  units  except  that 
for  quantity  are  identical.  "Whereas  the  electrostatic  unit  of  quantity 
is  determined  from  its  exerting  a  dyne  of  force  on  another  unit 
quantity,  the  electro-magnetic  unit  of  quantity  is  determined  from 
its  exerting  a  dyne  of  force,  when  moving  as  a  current,  on  a  unit 
magnetic  pole.  The  electro-magnetic  unit  is  about  3  x  10'"  times 
the  electrostatic  unit.  This  numerical  factor  is  the  same  as  the 
velocity  of  the  propagation  of  light  expressed  in  centimetres  per 
second.  This  fact,  combined  with  certain  mathematical  relations 
which  exist  between  the  two  units,  is  of  great  significance  in  sus- 
taining the  ether  theory  of  electricity. 

PRACTICAL  UNITS. 

Many  of  the  absolute  units  would  be  inconveniently  large  and 
others  would  be  inconveniently  small  for  practical  use.  Therefore 
the  following  units,  based  upon  the  electro-magnetic  units,  are  used : 

Electromotive  force Volt         =  108    electro-magnetic  units. 

Resistance Ohm        —  109  "  " 

Current Ampere  =  10~'  " 

Quantity Coulomb  —  10~  '  "  " 

Capacity Farad     =  lO"9  "  " 

Even  these  units  are  not  of  a  magnitude  suited  for  the  use  of  all 
electricians.  Thus  a  physician  uses  currents  whose  strengths  can 
be  more  easily  expressed  in  thousandths  of  an  ampere.  The  prefix 
milli-  is  therefore  used  for  "one  thousandth  "  and  a  milliampere  is 
the  thousandth  part  of  one  ampere.  Capacities  are  best  expressed 
in  millionths  of  a  farad  or  microfarads.  The  high  resistances 
offered  by  insulations  are  conveniently  expressed  in  megohms  = 
one  million  ohms. 


APPENDIX. 


APPLICATIONS    OF    THE    CALCULUS. 

I.  FALL  OF  BODIES. 

1.  Differential  Equations  for  Force  and  Motion.— These 

«re  three  in  number,  as  follows : 

i.    „  =  *.'. 

f  _dv  _a?  s 

J  ~  Tt  ~  Tf 

S.fds  =  vdv. 

These  equations  are  readily  derived  from  the  elementary  prin- 
ciples of  mechanics.  In  Art.  6  we  have  v  =  --.  Eeducing  the 
numerator  and  denominator  to  infinitesimals,  v  remains  finite,  and 

the  equation  becomes  v  =  -=- ,;  which  is  Equation  1st.     Therefore, 
cL  t 

if  the  space  described  by  a  body  is  regarded  as  a  function  of  the 
time,  the  first  differential  coefficient  expresses  the  velocity. 

Again  (Art.  12),  /  =  -,  where  /  represents  a  constant  force. 
t 

Making  velocity  and  time  infinitely  small,  we  get  the  intensity  of 
the  momentary  force,  /=  -^.  But,  by  Equation  1st,  v  =  ~ ; 

d*  s 
.-.£.—  -_ ;  which  is  Equation  2d.     Hence  we  learn  that  the  first 

differential  coefficient  of  the  velocity  as  a  function  of  the  time,  or 
the  second  differential  coefficient  of  the  space  as  a  function  of  the 
time,  expresses  the  force. 

Equation  3d  is  obtained  by  multiplying  the  1st  and  2d  cross- 
wise, and  removing  the  common  denominator. 

We  proceed  to  apply  these  equations  to  the  preparation  of  for- 
mulae for  falling  bodies. 

2.  Bodies   falling  through  Small  Distances  near  the 
Earth's  Surface. — In  this  case,  let  the  accelerating  force,  which 


454  APPENDIX. 

is  considered  constant,  be  called  g.  Then,  by  Eq.  2,g  =  ~^.  .-.  dv 
=  gdt.  Integrating,  we  have  v  =  g  t  +  C.  But,  since  v  =  0 
when  t  =  0,  .*.  v  =  g  t,  and  t  =  -,  as  in  Art.  27. 

Again,  substituting  g  t  for  v  in  Eq.  1,  ds  =  gtdt;  and  by 
integration,  s  =  $gF  +0;  but  C  =  0,  for  the  same  reason  as  be- 
fore ;  .*.  s  =  3  g  f,  and  £  =  \  — . 

Once  more,  equating  the  two  foregoing  values  of  t,  we  have 
t>  =  4/2  g  s,  and  s  =  =-. 

If,  in  the  equation,  s  =  %g  t1,  v  be  substituted  for  #  t,  we  have 
g  =  %vt,  or  vt  =  %  s;  that  is,  the  acquired  velocity  multiplied  by 
the  time  of  fall  gives  a  space  twice  as  great  as  that  fallen  through 
(Art.  21). 

3.  Bodies  falling  through  Great  Distances,        FIG.  1. 
so  that  Gravity  is  Variable,  according  to  the 

Law  in  Art.  16.— 

Suppose  a  body  to  fall  from  A  to  B  (Fig.  1),  to- 
ward the  centre  C.  Let  A  C=  a;  B  C  =  x;  D  C=r, 
the  radius  of  the  earth. 

The  force /at  B,  is  found  by  the  principle,  Art  16, 

*      »..       f—      t1  —      •    -> 
x 

4.  To  find  the  Acquired  Velocity. — Substitute  g  r*  or*  for 
/,  and  a  —  x  for  s,  in  Equation  3d,  and  we  have  g  r*  x~*  .d(a  —  x} 
=  vdv;    .'.  by  integration  |  v2  =  f  —  g  r*  ar*  d  x  =  g  r*  x~l  +C. 
But  v  =  0,  when  x  =  a ;  .'.  C  —  —  g  r*a~* ;  and 

£  v*  =  g  r*  x-1  -  g  r'  ar1 ; 


This  is  the  general  formula  for  the  acquired  velocity.    If  the 
body  falls  to  the  earth,  x  =  r,  and  the  formula  becomes 


FALL    OF    BODIES.  455 

Again,  if  the  body  falls  to  the  earth  through  so  small  a  space 
that  -  may  be  regarded  as  a  unit,  the  formula  reduces  to 


the  same  as  obtained  by  other  methods. 

If  a  body  falls  to  the  earth  from  an  infinite  distance,  it  does 
not  acquire  an  infinite  velocity.    For  then,  as  we  may  put  a  for 


(2 .  32^  .  3956  .  5280)7^  feet  =  6.95  miles. 

Therefore,  the  greatest  possible  velocity  acquired  in  falling  to 
the  earth  is  less  than  seven  miles ;  and  a  body  projected  upward 
with  that  velocity  would  never  return. 

5.  To  find  the  Time  of  Falling. — From  equation  first  we 

obtain  d  #=—  :  in  this,  substitute  d(a—x)  for  ds,  and     gr  ^a    x>* 

(a*)* 
for  v,  as  found  in  the  preceding  article ;  then 

,         (axy  .d(a  —  x)  _  /    a    \*    —  x~dx^ 

J 9  n  r*  ( n  T^H^         \£  (J  T  /       .  \-% 

\<o gr  \a  —  £)\  \a,  —  x) 

/.  by  integration  t  =  (^ j   .   I  —  x*  dx(a  —  *)""*. 

By  the  formula  in  the  calculus  for  reducing  the  index  of  x  we 
obtain 

/—  x'*d  x  (a  —  x}~*  =  (a  x  —  #5)3  —  -  vers~^  ( —  )  4-  G. 
%  \  a  / 

Now,  when  t  =  0,  x  =  a ;  .'.  C  =  -^  ; 

4 


6.  Bodies  falling  within  the  Earth  (sup-  FIG.  2. 

posed  to  be  of  uniform  density),  where 
G-ravity  Varies  as  the  Distance  from  the 
Centre. — 

Suppose  a  body  to  fall  from  A  to  B  (Fig.  2) ; 
and  let  D  C  =  r,  A  C  =  a,  and  B  C  =  x.    Then 

r:x::g:f=^x  =  force  at  B. 


456  APPENDIX. 

To  find  the  velocity  acquired.  —  By  Eq.  3d, 
vdv=fds-,  .-.  vdv=9  x.d(a-x)  =-. 


v*=  -    ~  +  G\  but  v  =  0  when  x  =  a; 


,.  C  =  f£  and  i  -  =  i<^>  ;.-.,=  {  «  („•-  «•)  }  * 


If  the  body  falls  from  the  surface  to  the  centre,  x  =  0,  and 

this  formula  becomes  r  =  (g  rfi  =  (32£  x  3956  x  5280)^  =  25,904 
feet  per  second. 

To  find  the  time  of  falling. — By  Equation  1st,  and  substitu- 

,.  .     ,.      ds      d(a—x)  dx  —dx 

tions,  we  obtain  d  t  —  —  =  — = = 

v  v  v 


—  dx  lr\%  T  —dx          /r\*      _.x 

—  -          -         -         ~  '  - 


_. 

=    ~    cos  '  -  +0 


When  f  =  0,x  =  a,-  =  l,  and  the  arc,  whose  cosine  is  1  =  0; 


x  cos     -. 

If  the  body  falls  to  the  centre,  x  =  0,  and  t  =  (- }    x  -  ;  in 

which  a  does  not  appear  at  all ;  so  that  the  time  of  falling  to  the 
centre  from  any  point  within  the  surface  is  the  same ;  and  equals. 

WMxMSOvV. -L  seconds,  or  2lm.5.88. 


IL  CENTRE  OF  GRAVITY. 


7.  Principle  of  Moments.— In  order  to  apply  the  processes 
of  the  calculus  to  the  determination  of  the  centre  of  gravity,  the 
principle  is  used,  which  was  proved  (Art.  78),  that  if  every  par- 
ticle of  a  body  be  multiplied  by  its  distance  from  a  plane,  and 
the  sum  of  the  products  be  divided  by  the  sum  of  the  particles, 
the  quotient  is  the  distance  of  the  common  centre  from  the  same 
plane.     The  product  of  any  particle  or  body  by  its  distance  from 
the  plane,  is  called  its  moment  with  respect  to  that  plane. 

8.  General  Formulae. — Let  B  A  C  (Fig.  3)  be  any  symmetri- 
cal curve,  having  A  JTfor  its  axis  of  abscissas,  and  A  Y,  at  right 


APPLICATION    OF    FORMULAS. 


FIG.  3. 


angles  to  it,  for  its  axis  of  ordinates.     It  is  obvious  that  the 

centre  of   gravity  of   the   line   BAG,  of   the  area  B  A  C,  of 

the  solid  of  revolution  around  the  axis 

A  X,  and  of  the  surface  of  the  same 

solid,  are  all  situated  on  A  X,  on  ac- 

count of  the  symmetry  of  the  figure. 

It  is  proposed  to  find  the  formula  for 

the  distance  of  the  centre  from  A  Y, 

in  each  of  these  cases.     Let  G  in  every 

instance  represent  the  distance  of  the 

general  centre  of  gravity  from  the  axis 

A  Y,  or  the  plane  A  Y,  at  right  angles  to  A  X.   The  distance  0 

would  plainly  be  the  same  for  the  half  figure  B  A  D,  as  for  the 

whole  B  A  C;   expressions  may  therefore  be  obtained  for  either, 

according  to  convenience. 

1.  The  line  A  B.  —  Let  x  be  the  abscissa,  and  y  the  ordinate  ; 

then  (dx1  +  dy*y~  is  the  differential  of  the  line  A  B.  For  brevity, 
let  s  =  the  line,  and  d  s  its  differential.  If  we  now  multiply  this 
differential  by  its  distance  from  A  Y,  x  d  s  is  the  moment  of  a 
minute  portion  of  the  line  ;  and  the  integral  of  it,  /  x  d  s,  is  the 
moment  of  the  whole.  Dividing  this  by  the  line  itself,  i.  e.  by  s, 


we  have 


fxds 


for  the  distance  G. 


2.  TJie  area  B  A  D. — The  differential  of  the  area  is  y  dx ;   the 
differential  of  its  moment  is  x  y  d  x ;   hence  the  moment  itself  is 

f  xy  dx:  and  the  distance  G  =  - — - — . 

area 

3.  The  solid  of  revolution. — The  differential  of  the  solid,  gen- 
erated by  the  revolution  of  A  B  on  A  X,  is  TT  y*d  x ;  the  differen- 
tial of  its  moment  is  TT  x  y'd  x ;  and  the  moment  is  f  n  x  y'd  x ; 

i.          j-i.    j-  j.          n       fTTxy'*dx 

hence  the  distance  G  = =£: . 

solid 

4.  The  surface  of  revolution. — The  differential  of  the  surface  is 
2  TT  y  d  s  ;   the  differential  of  its  moment  is  2  TT  x  y  d  s  ;  and  there- 
fore the  moment  isfZrrxyds',  and  the  distance  G  — '  — -— s. 


9.  Application  of  Formulae.— We  proceed  to  determine 
the  centre  of  gravity  in  a  few  cases  by  the  aid  of  these  formulas : 

1.  A  straight  line. — Imagine  the  line  placed  on  A  X,  with  one 
extremity  at  the  origin  A.  The  moment  of  a  minute  part  of  it  is 
v  d  x,  and  that  of  the  whole  is  ./'  x  d  x,  while  the  length  of  the 

whole  is  x;  .:  G  =  ^-^-  =  ^^—  =  $x,  as  it  evidently  should 


458  APPENDIX. 

be.     In  all  the  cases  considered  here,  C=  0,  because  the  function 
vanishes  when  x  does. 

2.  The  arc  of  a  circle. — By  formula  1st  we  have6r  —  ' •  but 

s 

d  s  =  (d  x*  +  difY  ;   by  the  equation  of  the  circle,  y*=  2  a  x  —  z* ; 


fxds        fx  adx  a  (*       xdx  a  ( 

.-. =  J  -  x  -  — -  =  -J  -  — -  =-  J  vers  1  x 

—  (2ax  —  x*Y  I  —  -  (s  —y)  =  a *'  =  a  — — ,  if  the  arc  is  dou- 

I  S  S  t 

bled  and  called  t,  and  c  (chord)  put  for  2  y.  Asa—-^  is  the  dis- 
tance from  the  origin  A,  and  a  =  radius  of  the  arc  ;  .•.  the  distance 
from  the  centre  of  the  circle  to  the  centre  of  gravity  of  the  arc, 

is  —,  which  is  a  fourth  proportional  to  the  arc,  the  chord,  and  the 

radius. 

When  the  arc  is  a  semi-circumference,  c  =  2  a,  and  t  =  TT  a ; 
.*.  the  distance  of  the  centre  of  gravity  of  a  semi-circumference 

from  the  centre  of  the  circle  is  — . 

7T 

3.  TJie  area  of  a  circular  sector. — Suppose  the  given  sector  to 
be  divided  into  an  infinite  number  of  sectors ;  then  each  may  be 
considered  a  triangle,  and  its  centre  of  gravity  therefore  distant 

from  the  centre  of  the  circle  by  the  line  -^-.    Hence  the  centres  of 

o 

gravity  of  all  the  sectors  lie  in  a  circular  arc,  whose  radius  is  —  ; 

so  that  the  centre  of  gravity  of  the  whole  sector  coincides  with 
the  centre  of  gravity  of  that  arc.  The  distance  of  the  centre  of 
gravity  of  the  arc  from  the  centre  of  the  circle,  by  the  preceding 


case,  is-  a  x  s  e  -*-  s  tf  =  -7-,  which  is  therefore  the  distance  of 
o          o          o  o  t 

the  centre  of  gravity  of  the^  sector  from  the  centre  of  the  circle. 

When  the  sector  is  a  semicircle  the  distance  becomes  —  ^  --- 

o  TT  a 


37T 


CENTRE    OF    GRAVITY  459 

4.  The  area  of  a  parabola.  —  The  equation  of  the  curve  is 

y*  =  p  x,  or  y  =  p~  z*  ; 
therefore  the  formula  2  for  moment, 

fxydx  =  fp*x^dx  =  f.p*  a£  (  +  (7  =  0)  ; 
but  the  area  of  the  half  parabola  =  f  p*  x*  ; 

...  G  =  lp*x%  -Hl>-^  =  \x. 

To  find  the  distance  of  the  centre  of  gravity  of  the  semi-parab- 
ola from  the  axis  A  X,  proceed  as  follows  :  The  differential  of  the 
area,  as  before,  equals  y  d  x  ;  and  the  distance  of  its  centre  from 
A  X  is  4  y-  Therefore  its  moment  with  respect  to  A  X  is  \  y*  d  x 
—  £  p  x  dx-,  and  the  moment  of  the  whole  is  f±p  x  dx  =  {  p  x9; 
/.  the  distance  of  the  centre  from 

AX  =  \px*  +  lp^xs  =  lp*x*^ly. 

5.  The  area  of  a  circular  segment.—  The  equation  of  the  circle 
is,  y  =  (2  a  x  —  x*)*.    Therefore  (formula  2), 

fxydx—fx(^ax  —  x*)?  d  x. 
Add  and  subtract  a  (2  a  x  —  x^  d  x,  and  it  becomes 

fa  (2  ax-  x^  d  x  -f(a  -x)(2ax  —  x*)?  dx  = 

(2ax—x't)?  (a—x)  dx 
^ 


~ 


When  x  =  a,  G  =  a  —  ^~  ;  and  the  distance  of  the  centre  of 

O  TT 

gravity  of  a  semicircle  from  the  centre  of  the  circle  =  ^.    When 

•x  =  2  a,  G  =  a,  as  it  plainly  should  be. 

6.  A  spherical  segment.  —  The  equation  of  the  circle  is  y1  — 
2ax-x\    Therefore  (formula  3), 


_ 


anx*  —  ±nx3~  IZa  —  ±x' 
When  x  —a,  G  =  f  a;  that  is,  the  centre  of  gravity  of  a  hem- 

isphere is  §  of  radius  from  the  surface,  or  |  of  radius  from  the 

centre  of  the  sphere.    If  x  =  2  a,  G  —  a. 

7.  A  right  cone.—  In  this  case  A  B  (Fig.  3),  is  a  straight  line, 

and  its  equation  is  y  =  a  x,  where  a  is  any  constant. 


460 


APPENDIX 


Hence  the  centre  of  gravity  of  a  cone  is  three-fourths  of  the  axis 
from  the  vertex.     See  Art.  75. 

8.  TJie  convex  surface  of  a  right  cone.  —  The  equation  is 

y  =  ax;  :.  df  =  tfdx*-,  and  (dx* 
Therefore  (formula  4), 


/2  TT  xy  ds= 

=  the  moment  of  the  surface.     The  surface  itself, 


The  centre  of  gravity  of  the  convex  surface  of  a  right  cone  is  on 
the  axis,  at  a  distance  equal  to  two-thirds  of  its  length  from  the 
vertex. 


III.  CENTKE  OF  OSCILLATION. 
10.   To  find  the  Moment  of  Inertia  of  a  Body  for  any 


given  Axis.  —  To  render  the  formula  I  = 


suitable  to  the 


application  of  the  calculus,  we  have  simply  to  substitute  the  sign 
of  integration  for  8,  and  d  M  for  m,  and  we  have 

m 


Mk 

It  is  useful  to  know  how  to  find  the  moment  of  inertia  with  respect 
to  any  axis  by  means  of  the  FIG.  4. 

known  moment  with  respect  to 
another  axis  parallel  to  it  and 
passing  through  the  centre  of 
gravity  of  the  body. 

Let  AZ  (Fig.  4)  be  the  axis 
passing  through  the  centre  of 
gravity  of  the  body  for  which 
the  moment  of  inertia  is  fr*dM, 
and  let  A'  Z'  be  the  axis  paral- 
lel to  it,  for  which  the  moment 
of  inertia,  ./>"  d  M  of  the  same 
mass  M,  is  to  be  determined. 
For  every  particle  m  of  the  body 
the  corresponding  value  of  A  m' 
is  ra  =  tf  +  ya.  In  like  man- 


CENTRE    OF     OSCILLATION.  461 

uer,  if  we  denote  the  co-ordinates  of  A'  by  a  and  0,  and  the  dis- 
tance between  the  axes  by  a,  we  shall  have  #3  —  a2  +  j3\  Now  the 
distance  of  the  particle  m  from  A'  Z'  is  r'2  =  (x  —  a)2+  (y  —  /3)* 
=  x*  +  tf  +a*  +  p  -2az-  2(3y  =  r*  +  a2  -  2  ax  -  2  p  y,  .-„ 
fr"*  dM=  fr*dM  +  cffd  M-2afxdM  -2(3fyd  M  =  a*  M 


since  A  Z  passes  through  the  centre  of  gravity  of  the  body.  Itence. 
the  moment  of  inertia  of  a  body  ivitli  respect  to  any  axis  is  equal  to 
the  moment  of  inertia  with  respect  to  a  parallel  axis  through  the 
centre  of  gravity,  plus  the  mass  of  the  body  multiplied  by  the  square- 
of  the  distance  between  the  two  axes. 

Put  C  =  the  moment  of  inertia  with  respect  to  an  axis  through 
the  centre  of  gravity  ;  then  the  distance  from  the  axis  of  suspen- 
sion to  the  eentre  of  oscillation,  the  axes  being  parallel,  will  be 


11.  Examples.  — 

1.  Find  the  centre  of  oscillation  of  a  slender  rod  or  straight 
line  suspended  at  any  point. 

Let  a  and  b  be  the  lengths  on  opposite  sides  of  the  axis  of  sus- 
pension, then  by  (1) 

_  fi*dM_          fr^dr         _  2  (a3  +  b3)  _  2(a*-ab  +  b9) 
:  '  Mk     ~  (a  +  b)$  (a-b)~3  (a*  -  V)  ~        3  (a  -  b) 

between  the  limits  r  =  +  a  and  r  =  —  b. 

If  the  rod  is  suspended  at  its  extremity,  5  =  0,  and  I  =  f  a.    If 
it  is  suspended  at  its  middle  point,  a  =  b  and  I  —  QC  . 

2.  Find  the  centre  of  oscillation  of  an  isosceles  triangle  vibra- 
ting about  an  axis  in  its  own  plane  passing  through  its  vertex. 

Put  b  and  h  for  the  base  and  altitude  of  the  triangle;  then  by 


If  the  axis  of  suspension  coincides  with  the  base  of  the  trian- 
gle, then  I  =  —   — — - — — —  —  =  -. 

3.  Find  the  centre  of  oscillation  of  a  circle  vibrating  about  an 

axis  in  its  own  plane. 

C  =»' dM=  2fx*  ydx  =  2  fa?  (£*  - 


x  = 


462  APPENDIX. 

Taking  this  integral  between  x  =  —  r  and  x  =  +  r,  we  have 

c=**-T  =  "-r- 

Substituting  this  value  of  C'in  (3)  we  have 


, 


4.  Find  the  centre  of  oscillation  of  a  cir- 
cle vibrating  about  an  axis  perpendicular 
to  it. 

Let  K  L  (Fig.  5)  be  an  elementary  ring 
whose  radius  is  x  and  whose  breadth  is  d  x  ; 
then 


Fro.  5. 


d  M  =  2 


TT  R4 


•xdx,  and  C  = 

7T  R* 


•  2  n  x  d  x 


„ 


r»  ni 

As  a  +  <r—  is  greater  than  a  +  —  -,  a  cir- 

cular pendulum  will  vibrate  faster  when  the 
axis  of  suspension  is  in  its  plane,  than  when 
it  is  perpendicular  to  it. 


IV.  CENTRE  OF  HYDROSTATIC  PRESSURE. 

12.  General  Formula. — Let  the  surface  pressed  upon  be 
plane  and  vertical ;  and  let  the  water  level  be  the  plane  of  refer- 
ence. Suppose  the  surface  to  have  a 
symmetrical  form  with  reference  to  a 
vertical  axis,  x,  whose  ordinate  is  y 
(Fig.  6).  A  horizontal  element  of  the 
surface  is  2  y  d  x,  and  (since  the  pres- 
sure varies  as  the  depth)  the  pressure 
on  that  element  2  x  y  d  x.  Hence  the 
whole  pressure  to  the  depth  x  is 
f%xydx=Zfxydx.  The  mo- 
ment of  the  pressure  on  the  element 

of  surface  is  2  x*  y  d  x ;   and  the  sum  of  all  the  moments  to  the 
same  -depth  is  /  2  z2  y  d  x  —  2  /  x*  y  d  x.     Therefore,  putting  p 

for  the  depth  of  the  centre  of  pressure,  p  =  -^ — — ,— . 

tf       fxydx 


FIG.  6. 


CENTRE    OF    HYDROSTATIC    PRESSURE.        463 

13.  Examples. 

1.  A  rectangle. — Let  its  height  =  h,  and  its  base  =  b  ;  then  2  y 
everywhere  equals  b,  and  a  horizontal  element  at  the  depth  x  is 
b  d  x,  the  pressure  on  it   is  b  x  d  x,   and  the   moment  of  that 
pressure  is  b  X*  d  x;  .:  the  depth  of  the  centre  of  pressure  p  = 
Cbx*dx      \bx3  +  c       0. 

^rr — T—  =  TT~ ~, — '•  Since  the  pressure  and  area  is  each  zero. 
f  bxdx  -^  bx  +  c 

when  x  is  zero,  c  and  c  both  disappear,  and  p  =  f  x,  which  for  the 
whole  surface  becomes  p  =  f  h.  That  is,  the  centre  of  pressure  on 
a  vertical  rectangular  surface  reaching  to  the  water  level,  is  two- 
thirds  of  the  distance  from  the  middle  of  the  upper  side  to  the 
middle  of  the  lower. 

2.  A  triangle  whose  vertex  is  at  the  surface  of  the  water,  and  its 
base  horizontal — Let  the  triangle  be  isosceles,  its  height  =  h,  and 

its  base  =  b',  then  h  :  b  : :  x  :  2  y  =  j-  x.   Therefore^?  =  -— T 


J  n 


x*dx 


=  * — 3  =  |  x ;   and  for  the  whole  height,  |  h. 

3   X 

If  the  triangle  is  not  isosceles,  it  may  be  easily  shown  that  the 
centre  of  pressure  is  on  the  line  joining  the  vertex  and  the  middle 
of  the  base,  at  a  distance  from  the  vertex  equal  to  three-fourths  of 
the  length  of  that  line. 

3.  A   triangle  whose  base  is  at  the  water  level. — Then  h  :  b 

•h  —  x  :  %  y  =  b  —  T  x.  Therefore  the  pressure  is  f—  bxdx 
— f  —  =r  x*  d  x,  because  d  x  is  negative.  The  moment  of  the 
pressure  is  /—  b  x3  d  x  —  f—  -= 

J  b  x  d  x  -\     i    ..vi*..  3  •"    i    11- 

Therefore  p  =  • 


.  and,  when  x  =  h,  «Ui  becomes  4  h. 

—  —  4z    ' 

In  general,  the  centre  of  pressure  is  at  the  middle  of  the  line  join- 
ing the  vertex  and  the  middle  of  the  base. 

4.  A  parabola  whose  vertex  is  at  the  surface. — As  y  =  p'2  z2, 

,       .             fx'p^x'dx      fx^dx      $x$      5  57    , 

therefore  p  =  - — ^ — ; =  - — =  — -  =  =a;;  or  -  h,  for 

fxv*x?dx  ?  1 

the  whole  area. 


464  APPENDIX. 

5.  A  parabola  whose  base  is  at  the  surface. — As  h  —  x  is  the 

—  f  (h  —  xYx~  d  x 
depth  of  an  element,  d  x  is  negative,    p  =  —  —  = 

-f(h  -x)  x^dx 

f  (h'x*  d  x  -  2  h  x%  d  x  +  x%dx)  _  |  h*x?  -  |  h  x?  +  %  &  _ 
f  (h  x*  d  x  -  x?  d  x)  |  h  x*  -  |  z* 

4  h*  —  \  h  x  +  7  v? 
— — ,5_  1 — - — ;   and  when  x  =  h,  the  expression  becomes 

jy-§y  +  ia'_4 

~ 


V.  ANGULAR  RADIUS  OF  THE  PRIMARY  AND  SECONDARY  RAIN- 
BOW AND  THE  HALO. 

14.  The  Primary  Rainbow. — Since  the  primary  bow  is 
formed  by  those  rays  which,  on  emerging  after  one  reflection, 
make  the  largest  angle  with  the  incident  rays,  proceed  to  find 
what  angle  of  incidence  will  cause  the  largest  deviation  of  the 
•emerging  rays. 

In  Fig.  7,  let  x  —  angle  of  inci-  FIG.  7. 

dence ;  y  —  angle  of  refraction ;  z  = 
angle  of  deviation ;  n  —  index  of  re- 
fraction. Then,  in  the  quadrilateral 
BDGK,  DBK=DGK=x-y, 
angle  at  D  =  360  -  2  y ;  .-.  K=  z  = 


x 

But  sin  x  =  n  sin  y; 


.:  cos  x  d  x  =  n  cos  y  d  y,  and  -     = 


. 
d  x      n  cos  y 

By  substitution, 


/.  2  cos  x  =  n  cos  y  ;  and  4  cos*  x  =  n*  cos*  y. 
But  sin7  x  =  n9  sin"  y  ; 

.*.  3  cos7  x  +  I  =  n"  ;  since  sin"  +  cos"  =  1. 

8 


Jtf~= 

.-.  cos  x  =  y  — — 


If  1.33  and  1.55,  the  values  of  n  for  extreme  red  and  violet,  be 
used  in  this  formula,  we  obtain  x,  and  therefore  y  and  z,  for  the 
limiting  angles  of  the  primary  bow. 


RADIUS  OF  RAINBOW  AND  HALO. 


465 


15.  The  Secondary  Bow.— TG  find  the  angle  of  minimum 
deviation.     Using  the  same  notation  as  before,  we  have  in  the 
pentagon  G  ED  BK  (Fig.  8),  G  =  B  = 
180  —  x  +  y;  E  —  D  =  2y;  .•.  K=z=. 

180  +  2x  -  6y; 


. 

dx  dx 

=  2  ;  and  3  cos  x  =  n  cos  y\ 


n  cos  y 

.:  9  cos*  x  =  n1  cos2  y  ; 

but     sin2  a?  =  ri1  sin"  y  ; 

.-.  8  cos'  x  +  I  =  n1  ; 


.'.   COS  X  = 


I3  —  I 


which,  as  before,  will  furnish  z  for  each  limiting  color  of  the  sec- 
ondary bow. 

16.  The  Common  Halo. — Let  D  E  (Fig.  9)  be  the  ray  from 
the  sun,  and  F G  the  emergent  ray.     Let  D Ep  —  x;  KEF  —  y^ 
KFE  =  x'-,GFp  =  yr;  1= 
z  =  x  —  y  +  y'  —  x'.      Now,                           FlG-  9- 
•a  +  x'  =  »'  KF  -  C  =  60°.  4 J? 


sin  x  =  n  sin  y, 
and  siny'=  n  sin  x' ; 

.*.  x  =  sin"1  (n  sin  y), 
und       y'  =  sin"1  (n  sin  x')  = 

sin"1  jw  sin  ((7 — y)}. 
By  substitution, 

z  =  sin"1  (w  sin  y)  +  sin"1  { w  sin  ( C  —  y} }  —  C.  Therefore  z  is  a 
function  of  y ;  and,  by  differentiating,  we  have 

d  z  _        n  cos  y  n  cos  (C  —  y}         _ 

dy~  Vl  —  n*~sm*  y       Vl  -  n*  sin2  (C  —  y)~ 

ri1  cos2  y  n*  cos2  ( 0  —  y) 

"  I  —  H*  sin2  y  ~  1  —  w2  sin2  ( C  —  y) ' 
1  —  sin"  y          1  —  sin"  (C  —  y)  m 
'*'  1  -  ws'sTnTy  ~  1  -  rf  shi2((7-.y) ' 
.-.  (ws  —  1)  sin1  y=(tf-  1)  sin2  (C  -  y) j 
.*.  y  =  C  —  y,  and  y  =  £  (7; 
and  *'  =  J  (7. 

Hence,  the  minimum  deviation  occurs  when  the  ray  within  the 
crystal  is  equally  inclined  to  the  sides.  Knowing  n,  the  index  of 
refraction  for  ice,  x,  and  its  equal,  y',  can  be  obtained,  and  then  z, 
the  deviation  required. 


OLMSTED'S  COLLEGE  ASTRONOMY. 


Third  Stereotype  Edition,  Revised,  with  Additions  by  Coffin, 


An  Introduction  to  Astronomy  for  the  Use  of  Students  in  College.  By 
DENISON  OLMSTED,  LL.D.,  Professor  of  Astronomy  in  Yale  College,  and 
E.  S.  SNELL,  LL.D.,  Professor  of  Mathematics  in  Amherst  College.  Third 
Edition,  revised,  with  additions  by  Prof.  SELDEN  J.  COFFIN,  Lafayette 
College.  Octavo,  cloth,  pp.  viii.,  236,  with  numerous  illustrations  and 
diagrams.  Price,  for  introduction,  $1.60  ;  by  mail,  $1-75. 

The  subject  of  Spectrum  Analysis  appearing  to  demand  more  extended  treatment, 
Chapter  XX.  has  been  prepared  for  this  edition,  which  also  contains  a  number  of 
revisions  and  corrections  in  accordance  with  the  best  authorities.  A  few  Articles  have 
been  rewritten,  and  some  matter  added. 

The  special  advantages  recognized  in  this  text-book  are  :  First — Absence  of  all 
superfluous  matter,  and  its  moderate  size,  rendering  its  contents  capable  of  being 
compassed  within  the  limited  time  generally  allotted  to  the  subject.  Second — Its  dis- 
tinct and  accurate  mathematical  demonstrations,  showing  the  strictly  scientific  basis 
and  methods  of  Astronomical  work,  and  rendering  the  book  valuable,  both  as  a  means 
of  mental  discipline  and  as  an  introduction  to  the  practical  application  of  its  principles. 

Copies  for  examination  mailed  to  Teachers  and  Professors  upon  receipt  of  One 
Dollar. 


EXTRACTS    FROM    NOTICES. 

From  Prof.  C.  C.  Ferguson,  Tennessee  Valley  College.  Evansville,  April  2,  1891. 

An  examination  of  the  Astronomy  convinced  me  of  its  merits,  and  we  desire  to  introduce 

the  work  into  our  regular  course. 

From  Prof.  W.  C.   Bartol,  Bucknell  University.  Lewisburg,  Pa.,  Dec.  20,  1890. 

I  like  it  and  will  use  it  in  my  class  next  term. 

From  Prof.  W.  A.   Rogers,  Colby  University.  Waterville,  Me.,  Dec.  20,  1890. 

I  have  decided  to  adopt  the  Astronomy. 

From  Prof.  J.  M.  Taylor,  Washington  University.  Seattle,  June  23,  1890. 

I  have  carefully  examined  the  Astronomy  and  am  much  pleased  with  it.     I  have  adopted  i' 

as  the  text-book  for  my  classes. 

From  Prof.  John  W.  Caldwell,  Tulane  University.  New  Orleans,  Aug.  i,  1889 

I  have  carefully  examined  it  and  have  concluded  to  adopt  it. 


I  I   I  I   I  I  I;  I 


A 


>A  j  v*:c.! 

S        >:K  c  0  §  V  I  )V\  BO  «• 


• 


;, 


\r^ 


8  l 


<;    i 
g    3M 


666"  94l""l24 o"        OJITViHO^ 


I  I 


1  I 

i 


f!!!t 
g  |(iCl 

3  s 


t-UBBARYO/- 


(flF-CALIFOBt! 

~*^.    * 


,     a 

\    I 


%133NVS01^ 
'^         ^\\E-UNIVER^ 


i<\t-lllBRARY/)> 


